:= PolynomialRing(RationalField(), 6);
> B := [
> a + b + c + d + e + f,
> a*b + b*c + c*d + d*e + e*f + f*a,
> a*b*c + b*c*d + c*d*e + d*e*f + e*f*a + f*a*b,
> a*b*c*d + b*c*d*e + c*d*e*f + d*e*f*a + e*f*a*b + f*a*b*c,
> a*b*c*d*e + b*c*d*e*f + c*d*e*f*a + d*e*f*a*b +
> e*f*a*b*c + f*a*b*c*d,
> a*b*c*d*e*f - 1];
> I := ideal;
> time Groebner(I);
Time: 1.459
> A := AlgebraicClosure();
> time V := Variety(I, A);
Time: 4.219
> #V;
156
```
We now notice that there are 28 variables in $A$ and we check that all elements of $V$ satisfy the original polynomials.
```magma
> Rank(A);
30
> V[1];
<-1, -1, -1, -1, r1 + 4, -r1>
> V[156];
> {Evaluate(f, v): v in V, f in B};
{
0
}
```
We now simplify $A$ to ensure that it represents a true field, and prune away useless variables now having linear defining polynomials.
```magma
> time Simplify(A);
Time: 3.330
> Prune(A);
> A;
Algebraically closed field with 3 variables
Defining relations:
[
r3^2 - 1/3*r3*r2*r1 - 5/3*r3*r2 + 2/3*r3*r1 - 2/3*r3 + r2*r1 + 4*r2 + 1,
r2^2 - r2*r1 - 4*r1 - 1,
r1^2 + 4*r1 + 1
]
> V[1];
<-1, -1, -1, -1, r1 + 4, -r1>
> V[156];
<2/3*r3*r2*r1 + 7/3*r3*r2 - 1/3*r3*r1 - 2/3*r3 + 5/3*r2*r1 +
19/3*r2 + 2/3*r1 + 4/3, -2/3*r3*r2*r1 - 7/3*r3*r2 + 1/3*r3*r1
+ 2/3*r3 - 5/3*r2*r1 - 19/3*r2 - 2/3*r1 - 4/3, -4/3*r2*r1 -
14/3*r2 - 1/3*r1 - 2/3, 2/3*r3*r2*r1 + 7/3*r3*r2 - 1/3*r3*r1
- 2/3*r3 - 5/3*r2*r1 - 19/3*r2 + 1/3*r1 - 1/3, -2/3*r3*r2*r1
- 7/3*r3*r2 + 1/3*r3*r1 + 2/3*r3 + 5/3*r2*r1 + 19/3*r2 -
1/3*r1 + 1/3, 4/3*r2*r1 + 14/3*r2 + 1/3*r1 + 2/3>
```
Finally we compute an absolute polynomial for $A$, and then modify $A$ in place using [`Absolutize`](https://docs.magma-maths.org/GlobalFields/AlgebraicallyClosedFields/absolute.md#procedure-fldac-absolutize) to make $A$ be defined by one polynomial of degree 8.
```magma
> time AbsolutePolynomial(A);
x^8 + 4*x^6 - 6*x^4 + 4*x^2 + 1
Time: 0.080
> time Absolutize(A);
Time: 0.259
> A;
Algebraically closed field with 1 variables
Defining relations:
[
r1^8 + 4*r1^6 - 6*r1^4 + 4*r1^2 + 1
]
> V[1];
<-1, -1, -1, -1, 1/2*r1^6 + 2*r1^4 - 7/2*r1^2 + 3, -1/2*r1^6 -
2*r1^4 + 7/2*r1^2 + 1>
> V[156];
> {Evaluate(f, v): v in V, f in B};
{
0
}
```
#### `Example: Splitting Field of a Degree 8 Polynomial (ex-862d19)`
In this example we compute the splitting field of a certain polynomial of degree 8.
We first set $f$ to a degree-8 polynomial using the database of polynomials with given Galois group. The Galois group has order 16, so we know that the splitting field will have absolute degree 16.
```magma
> P := PolynomialRing(IntegerRing());
> load galpols;
Loading "/home/magma/libs/galpols/galpols"
> PolynomialWithGaloisGroup(8, 6);
x^8 - 2*x^7 - 9*x^6 + 10*x^5 + 22*x^4 - 14*x^3 - 15*x^2 + 2*x + 1
> f := $1;
> #GaloisGroup(f);
16
```
We next create an algebraic closure $A$ and compute the roots of $f$ over $A$.
```magma
> A := AlgebraicClosure();
> r := Roots(f, A);
> #r;
8
> A;
Algebraically closed field with 8 variables
Defining relations:
[
r8^8 - 2*r8^7 - 9*r8^6 + 10*r8^5 + 22*r8^4 - 14*r8^3 - 15*r8^2 + 2*r8 + 1,
r7^8 - 2*r7^7 - 9*r7^6 + 10*r7^5 + 22*r7^4 - 14*r7^3 - 15*r7^2 + 2*r7 + 1,
r6^8 - 2*r6^7 - 9*r6^6 + 10*r6^5 + 22*r6^4 - 14*r6^3 - 15*r6^2 + 2*r6 + 1,
r5^8 - 2*r5^7 - 9*r5^6 + 10*r5^5 + 22*r5^4 - 14*r5^3 - 15*r5^2 + 2*r5 + 1,
r4^8 - 2*r4^7 - 9*r4^6 + 10*r4^5 + 22*r4^4 - 14*r4^3 - 15*r4^2 + 2*r4 + 1,
r3^8 - 2*r3^7 - 9*r3^6 + 10*r3^5 + 22*r3^4 - 14*r3^3 - 15*r3^2 + 2*r3 + 1,
r2^8 - 2*r2^7 - 9*r2^6 + 10*r2^5 + 22*r2^4 - 14*r2^3 - 15*r2^2 + 2*r2 + 1,
r1^8 - 2*r1^7 - 9*r1^6 + 10*r1^5 + 22*r1^4 - 14*r1^3 - 15*r1^2 + 2*r1 + 1
]
```
Finally we simplify $A$. There are defining polynomials of degrees 2 and 8 in the simplified field. The absolute polynomial of degree 16 defines the splitting field of $f$.
```magma
> time Simplify(A);
Time: 2.870
> A;
Algebraically closed field with 8 variables
Defining relations:
[
r8 + 1/2*r3*r1^6 - 2*r3*r1^5 - r3*r1^4 + 8*r3*r1^3 - 2*r3*r1^2 -
5*r3*r1 - 1/2*r3 + r1^7 - 3/2*r1^6 - 9*r1^5 + 4*r1^4 + 19*r1^3
- r1^2 - 9*r1 - 5/2,
r7 - 1/2*r3*r1^6 + 2*r3*r1^5 + r3*r1^4 - 8*r3*r1^3 + 2*r3*r1^2 +
5*r3*r1 + 1/2*r3,
r6 + r3 - 3/2*r1^7 + 2*r1^6 + 14*r1^5 - 5*r1^4 - 28*r1^3 + 3*r1^2
+ 19/2*r1,
r5 - 1/2*r3*r1^6 + 2*r3*r1^5 + r3*r1^4 - 8*r3*r1^3 + 2*r3*r1^2 +
4*r3*r1 + 1/2*r3 + r1^6 - r1^5 - 9*r1^4 - r1^3 + 14*r1^2 +
6*r1,
r4 + 1/2*r3*r1^6 - 2*r3*r1^5 - r3*r1^4 + 8*r3*r1^3 - 2*r3*r1^2 -
4*r3*r1 - 1/2*r3 - 1,
r3^2 - 3/2*r3*r1^7 + 2*r3*r1^6 + 14*r3*r1^5 - 5*r3*r1^4 -
28*r3*r1^3 + 3*r3*r1^2 + 19/2*r3*r1 + 3/2*r1^6 - r1^5 -
15*r1^4 - 4*r1^3 + 27*r1^2 + 11*r1 - 9/2,
r2 + 1/2*r1^7 - 3/2*r1^6 - 4*r1^5 + 10*r1^4 + 10*r1^3 - 16*r1^2 -
11/2*r1 + 3/2,
r1^8 - 2*r1^7 - 9*r1^6 + 10*r1^5 + 22*r1^4 - 14*r1^3 - 15*r1^2 +
2*r1 + 1
]
> AbsolutePolynomial(A);
x^16 - 36*x^14 + 488*x^12 - 3186*x^10 + 10920*x^8 - 19804*x^6 + 17801*x^4 -
6264*x^2 + 64
```
#### `Example: Splitting Field of a Quartic as a Number Field (ex-d2e9ca)`
In this example we present the splitting field of $x^4 - x - 1$, whose Galois group is $S_4$, as a number field.
```magma
> P := PolynomialRing(RationalField());
> f := x^4 - x - 1;
> #GaloisGroup(f);
24
> A := AlgebraicClosure();
> r := Roots(f, A);
> #r;
4
```
By default `NumberField` presents the splitting field as a tower, with one small defining polynomial per level: a root of $f$, a root of the residual cubic above it, then a root of the residual quadratic. The map $m$ sends the roots of $f$ in $A$ to roots of $f$ in $N$.
```magma
> N, m := NumberField(A);
> AbsoluteDegree(N);
24
> DefiningPolynomial(N);
$.1^2 + (r2 + r1)*$.1 + r2^2 + r1*r2 + r1^2
> DefiningPolynomial(BaseField(N));
$.1^3 + r1*$.1^2 + r1^2*$.1 + r1^3 - 1
> DefiningPolynomial(BaseField(BaseField(N)));
x^4 - x - 1
> Evaluate(f, m(r[1][1]));
0
```
With `Absolute` the same field is defined by a single polynomial of degree 24 in a primitive element. The size of that polynomial grows quickly with the degree of the field, which is why the tower is the default.
```magma
> Na, ma := NumberField(A : Absolute);
> DefiningPolynomial(Na);
x^24 + 80*x^20 + 340*x^18 + 7520*x^16 - 23120*x^14 + 1241022*x^12 -
462400*x^10 + 48052400*x^8 - 61067260*x^6 + 756753632*x^4 +
2074608720*x^2 + 1215289321
> Evaluate(f, ma(r[1][1]));
0
```
#### `Example: Splitting Field of Degree 720 as a Tower (ex-fe6f00)`
For larger splitting fields the tower is the only practical presentation. The splitting field of $x^6 - x - 1$, whose Galois group is $S_6$, has degree 720. As a tower with one extension for each variable of $A$ it takes the time shown below. The `Absolute` presentation is orders of magnitude more expensive, and its defining polynomial is far too large to print.
```magma
> P := PolynomialRing(IntegerRing());
> f := x^6 - x - 1;
> #GaloisGroup(f);
720
> A := AlgebraicClosure();
> r := Roots(f, A);
> time N, m := NumberField(A);
Time: 20.140
> Degree(N), AbsoluteDegree(N);
2 720
> Evaluate(f, m(r[1][1]));
0
```
## Rational Function Fields
- [Introduction](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/introduction.md)
- [Creation Functions](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md)
- [Creation of Structures](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#creation-of-structures)
- [`FunctionField(R): Rng → FldFunRat`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-functionfield-rng)
- [`RationalFunctionField(R): Rng → FldFunRat`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-rationalfunctionfield-rng)
- [`FunctionField(R, r): Rng, RngIntElt → FldFunRat`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-functionfield-rng-rngintelt)
- [`RationalFunctionField(R, r): Rng, RngIntElt → FldFunRat`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-rationalfunctionfield-rng-rngintelt)
- [`FieldOfFractions(P): RngUPol → FldFunRat`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-fieldoffractions-rngupol)
- [Names](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#names)
- [`AssignNames(~F, s): FldFunRat, [ MonStgElt ]`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#procedure-assignnames-fldfunrat-monstgelt-ref)
- [`Name(F, i): FldFunRat, RngIntElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-name-fldfunrat-rngintelt)
- [Creation of Elements](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#creation-of-elements)
- [`F ! [a, b]: FldFunRat, RngUPolElt, RngUPolElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#operation-op-fldfunrat-rngupolelt-rngupolelt)
- [`elt< F | a, b >: FldFunRat, RngUPolElt, RngUPolElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#constructor-constructor-elt-fldfunrat-rngupolelt-rngupolelt-fldfunratelt)
- [`F ! a: FldFunRat, FldElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#operation-op-fldfunrat-fldelt)
- [`K . i: FldFunRat, RngIntElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#operation-operation-fldfunrat-rngintelt-fldfunratelt)
- [`One(F): FldFunRat → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-one-fldfunrat)
- [`Identity(F): FldFunRat → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-identity-fldfunrat)
- [`Zero(F): FldFunRat → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-zero-fldfunrat)
- [`Representative(F): FldFunRat → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#function-representative-fldfunrat)
- [Example: Creating Rational Function Field Elements](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/creation.md#example-ex-64b44d)
- [Structure Operations](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md)
- [Related Structures](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#related-structures)
- [`IntegerRing(F): FldFunRat → RngPol`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-integerring-fldfunrat)
- [`RingOfIntegers(F): FldFunRat → RngPol`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-ringofintegers-fldfunrat)
- [`BaseRing(F): FldFunRat → Rng`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-basering-fldfunrat)
- [`CoefficientRing(F): FldFunRat → Rng`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-coefficientring-fldfunrat)
- [`Rank(F): FldFunRat → RngIntElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-rank-fldfunrat)
- [`ValuationRing(F): FldFunRat → RngVal`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-valuationring-fldfunrat)
- [`ValuationRing(F, f): FldFunRat, RngUPolElt → RngVal`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-valuationring-fldfunrat-rngupolelt)
- [`Category(R): FldFunRat → Cat`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-category-fldfunrat)
- [`Parent(R): FldFunRat → Pow`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-parent-fldfunrat)
- [`PrimeRing(R): FldFunRat → Rng`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-primering-fldfunrat)
- [Invariants](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#invariants)
- [`Characteristic(F): FldFunRat → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-characteristic-fldfunrat)
- [Ring Predicates and Booleans](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#ring-predicates-and-booleans)
- [`IsCommutative(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-iscommutative-fldfunrat)
- [`IsUnitary(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isunitary-fldfunrat)
- [`IsFinite(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isfinite-fldfunrat)
- [`IsOrdered(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isordered-fldfunrat)
- [`IsField(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isfield-fldfunrat)
- [`IsEuclideanDomain(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-iseuclideandomain-fldfunrat)
- [`IsPID(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-ispid-fldfunrat)
- [`IsUFD(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isufd-fldfunrat)
- [`IsDivisionRing(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isdivisionring-fldfunrat)
- [`IsEuclideanRing(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-iseuclideanring-fldfunrat)
- [`IsPrincipalIdealRing(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isprincipalidealring-fldfunrat)
- [`IsDomain(F): FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#function-isdomain-fldfunrat)
- [`F eq G: FldFunRat, Rng → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#operation-op-eq-fldfunrat-rng)
- [`F ne G: FldFunRat, Rng → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#operation-op-ne-fldfunrat-rng)
- [Homomorphisms](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#homomorphisms)
- [`hom< P -> S | f, y₁, ..., yₙ >: FldFunRat, Rng → Map`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#constructor-constructor-hom-fldfunrat-rng-map)
- [`hom< P -> S | y₁, ..., yₙ >: FldFunRat, Rng → Map`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#constructor-constructor-hom-fldfunrat-rng-map-2)
- [Example: Homomorphism to a Number Field](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/operation.md#example-ex-bca4bf)
- [Element Operations](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md)
- [Arithmetic](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#arithmetic)
- [`+ a: FldFunRatElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-operation-fldfunratelt-fldfunratelt)
- [`- a: FldFunRatElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-operation-fldfunratelt-fldfunratelt-2)
- [`a + b: FldFunRatElt, FldFunRatElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-plus-fldfunratelt-fldfunratelt)
- [`a - b: FldFunRatElt, FldFunRatElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-minus-fldfunratelt-fldfunratelt)
- [`a * b: FldFunRatElt, FldFunRatElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-times-fldfunratelt-fldfunratelt)
- [`a / b: FldFunRatElt, FldFunRatElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-div-fldfunratelt-fldfunratelt)
- [`a ^ k: FldFunRatElt, RngIntElt → FldFunRatElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-pow-fldfunratelt-rngintelt)
- [Equality and Membership](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#equality-and-membership)
- [`a eq b: FldFunRatElt, FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-eq-fldfunratelt-fldfunratelt)
- [`a ne b: FldFunRatElt, FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-ne-fldfunratelt-fldfunratelt)
- [`a in F: FldFunRatElt, FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-in-fldfunratelt-fldfunrat)
- [`a notin F: FldFunRatElt, FldFunRat → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#operation-op-notin-fldfunratelt-fldfunrat)
- [Numerator, Denominator, and Degree](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#numerator-denominator-and-degree)
- [`Numerator(f): FldFunRatElt → RngElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-numerator-fldfunratelt)
- [`Numerator(f): RngUPolElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-numerator-rngupolelt)
- [`Numerator(f): RngMPolElt → RngMPolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-numerator-rngmpolelt)
- [`Denominator(f): FldFunRatElt → RngElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-denominator-fldfunratelt)
- [`Denominator(f): RngUPolElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-denominator-rngupolelt)
- [`Denominator(f): RngMPolElt → RngMPolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-denominator-rngmpolelt)
- [`Degree(f): FldFunRatElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-degree-fldfunratelt)
- [`TotalDegree(f): FldFunRatElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-totaldegree-fldfunratelt)
- [`WeightedDegree(f): FldFunRatElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-weighteddegree-fldfunratelt)
- [`Numerator(f, R): FldFunRatElt → RngElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-numerator-fldfunratelt-2)
- [`Denominator(f, R): FldFunRatElt → RngElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-denominator-fldfunratelt-2)
- [Predicates on Ring Elements](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#predicates-on-ring-elements)
- [`IsZero(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-iszero-fldfunratelt)
- [`IsOne(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-isone-fldfunratelt)
- [`IsMinusOne(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-isminusone-fldfunratelt)
- [`IsNilpotent(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-isnilpotent-fldfunratelt)
- [`IsIdempotent(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-isidempotent-fldfunratelt)
- [`IsUnit(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-isunit-fldfunratelt)
- [`IsZeroDivisor(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-iszerodivisor-fldfunratelt)
- [`IsRegular(a): FldFunRatElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-isregular-fldfunratelt)
- [Evaluation](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#evaluation)
- [`Evaluate(f, r): FldFunRatUElt, RngElt → FldFunRatUElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-evaluate-fldfunratuelt-rngelt)
- [`Evaluate(f, v, r): FldFunRatMElt, RngIntElt, RngElt → FldFunRatMElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-evaluate-fldfunratmelt-rngintelt-rngelt)
- [`Evaluate(f, S): FldFunRatMElt, [RngElt] → RngElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-evaluate-fldfunratmelt-rngelt)
- [Decomposition](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#decomposition)
- [`Decomposition(f): FldFunRatUElt → [[FldFunRatUElt]]`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-decomposition-fldfunratuelt)
- [Example: Decomposition of a Rational Function](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#example-ex-221d21)
- [Derivative](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#derivative)
- [`Derivative(f): FldFunRatUElt → FldFunRatUElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-derivative-fldfunratuelt)
- [`Derivative(f, k): FldFunRatUElt, RngIntElt → FldFunRatUElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-derivative-fldfunratuelt-rngintelt)
- [`Derivative(f, v): FldFunRatMElt, RngIntElt → FldFunRatMElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-derivative-fldfunratmelt-rngintelt)
- [`Derivative(f, v, k): FldFunRatMElt, RngIntElt, RngIntElt → FldFunRatMElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-derivative-fldfunratmelt-rngintelt-rngintelt)
- [Partial Fraction Decomposition](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#partial-fraction-decomposition)
- [`PartialFractionDecomposition(f): FldFunRatUElt → [ ]`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-partialfractiondecomposition-fldfunratuelt)
- [`SquarefreePartialFractionDecomposition(f): FldFunRatUElt → [ ]`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#function-squarefreepartialfractiondecomposition-fldfunratuelt)
- [Example: Partial Fraction Decomposition](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/element.md#example-ex-2ba727)
- [Padé-Hermite Approximants](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md)
- [Introduction](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#introduction)
- [Ordering of Sequences](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#ordering-of-sequences)
- [`MaximumDegree(f): SeqEnum → RngIntElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#function-maximumdegree-seqenum)
- [Example: Maximum Degree of a Sequence](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-ea74e8)
- [`TypeOfSequence(f): SeqEnum → RngIntElt, RngIntElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#function-typeofsequence-seqenum)
- [Example: Type of a Sequence](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-8d75ca)
- [`MinimalVectorSequence(f,n): SeqEnum, RngIntElt → SeqEnum`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#function-minimalvectorsequence-seqenum-rngintelt)
- [Example: Minimal Vector Sequence of Power Series](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-67b928)
- [Example: Minimal Vector Sequence of Polynomials](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-d1c2b6)
- [Example: Minimal Vector Sequence of Higher Order](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-328015)
- [Example: Minimal Vector Sequence with Distortion](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-1d53c0)
- [Approximants](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#approximants)
- [`PadeHermiteApproximant(f,d): SeqEnum, SeqEnum → ModTupRngElt, SeqEnum, RngIntElt`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#function-padhermapp)
- [Example: Pade-Hermite Approximants](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-05a6f0)
- [Example: Pade-Hermite Approximant with Distortion](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-e98d49)
- [Example: Degrees and Valuations of Approximants](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-ea09f6)
- [`PadeHermiteApproximant(f,m): SeqEnum, RngIntElt → ModTupRngElt, SeqEnum`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#function-padehermiteapproximant-seqenum-rngintelt)
- [Example: Pade-Hermite Approximants of Vectors](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-b5d830)
- [Example: Approximants of Sine, Cosine, and Exponential](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#example-ex-15b388)
- [Bibliography](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/bibliography.md)
### Introduction
Given a ring $R$ such that there is a greatest-common-divisor algorithm for polynomials over $R$, Magma allows the construction of a rational function field $K$ in any number of indeterminates over $R$. Such function fields are objects of type `FldFunRat` with elements of type `FldFunRatElt`. The elements of $K$ are fractions whose numerators and denominators lie in the corresponding polynomial ring over $R$. As for polynomial rings, the different univariate and multivariate cases are distinguished, since the fractions just use the different representations given by the different cases of polynomial rings. A fraction $f$ lying in a function field $K$ is always *reduced*; this means that the numerator and denominator of $f$ are coprime and the denominator of $f$ is *normalized* (monic over fields and positive over ${\mathbb{Z}}$). Note that $R$ itself need not be a field. Thus it is possible, for example, to create the rational function field $K = {\mathbb{Z}}(t)$ which is mathematically equal to ${\mathbb{Q}}(t)$ of course, but will be represented slightly differently. A fraction in ${\mathbb{Q}}(t)$ will have a monic denominator (and the coefficients of both the numerator and denominator may be non-integral), while a fraction in ${\mathbb{Z}}(t)$ will have a positive denominator (and the coefficients of both the numerator and denominator will be integral). Thus the fractions $(3t + 2)/(4t - 2) \in {\mathbb{Z}}(t)$ and $((3/4)t + 1/2)/(t - 1/2) \in {\mathbb{Q}}(t)$ are equal and are both reduced in their respective fields. It is generally much better to use the domain of integers instead of the field of fractions for the coefficient ring $R$ (so it is better to use ${\mathbb{Z}}(t)$ instead of ${\mathbb{Q}}(t)$) since arithmetic is much faster, but the use of a field of fractions for the coefficient ring may be more desirable for output purposes.
### Creation Functions
#### Creation of Structures
##### `FunctionField(R): Rng -> FldFunRat`
##### `RationalFunctionField(R): Rng -> FldFunRat`
```magma
Global: BoolElt Default: true
```
Create the field ${\mathbb{F}}$ of rational functions in $1$ indeterminate (consisting of quotients of univariate polynomials) over the integral domain $R$. The angle bracket notation may be used to assign names to the indeterminates, just as in the case of polynomial rings, e.g.: `K := FunctionField(IntegerRing());`. By default, the unique *global* univariate function field over $R$ will be returned; if the parameter `Global` is set to `false`, then a non-global univariate function field over $R$ will be returned (to which a separate name for the indeterminate can be assigned).
##### `FunctionField(R, r): Rng, RngIntElt -> FldFunRat`
##### `RationalFunctionField(R, r): Rng, RngIntElt -> FldFunRat`
```magma
Global: BoolElt Default: false
```
Create the field ${\mathbb{F}}$ of rational functions in $r$ indeterminates over the integral domain $R$. may be used to assign names to the indeterminates, just as in the case of polynomial rings, e.g.: `K := FunctionField(IntegerRing(), 3);`. By default, a *non-global* function field will be returned; if the parameter `Global` is set to `true`, then the unique global function field over $R$ with $n$ variables will be returned. This may be useful in some contexts, but a non-global result is returned by default since one often wishes to have several function fields with the same numbers of variables but with different variable names (and create mappings between them, for example). Explicit coercion is always allowed between function fields having the same number of variables (and suitable base rings), whether they are global or not, and the coercion maps the $i$-variable of one function field to the $i$-th variable of the other function field.
##### `FieldOfFractions(P): RngUPol -> FldFunRat`
Given a polynomial ring $P$, return its field of fractions $F$, consisting of quotients $f/g$, with $f, g\in P$. The angle bracket notation may be used to assign names to the indeterminates, just as in the case of polynomial rings: `K := FieldOfFractions(P);`. If this function is called more than once for a fixed $P$, then the identical function field will be returned each time.
#### Names
##### `AssignNames(~F, s): FldFunRat, [ MonStgElt ]`
Procedure to change the name of the indeterminates of a function field $F$. The $i$-th indeterminate will be given the name of the $i$-th element of the sequence of strings $s$ (for $1\leq i\leq \#s$); the sequence may have length less than the number of indeterminates of $F$, in which case the remaining indeterminate names remain unchanged. This procedure only changes the name used in printing the elements of $F$. It does *not* assign to identifiers corresponding to the strings the indeterminates in $F$; to do this, use an assignment statement, or use angle brackets when creating the field. Note that since this is a procedure that modifies $F$, it is necessary to have a reference `~F` to $F$ in the call to this function.
##### `Name(F, i): FldFunRat, RngIntElt -> FldFunRatElt`
Given a function field $F$, return the $i$-th indeterminate of $F$ (as an element of $F$).
#### Creation of Elements
##### `F ! [a, b]: FldFunRat, RngUPolElt, RngUPolElt -> FldFunRatElt`
##### `elt< F | a, b >: FldFunRat, RngUPolElt, RngUPolElt -> FldFunRatElt`
Given the rational function field $F$ (which is the field of fractions of the polynomial ring $R$), and polynomials $a, b$ in $R$ (with $b\neq 0$), construct the rational function $a / b$.
##### `F ! a: FldFunRat, FldElt -> FldFunRatElt`
Given the rational function field $F$ as a field of fractions of $R$, and a polynomial $a\in R$, create the rational function $a=a/1$ in $F$.
##### `K . i: FldFunRat, RngIntElt -> FldFunRatElt`
The $i$-th generator for the field of fractions $K$ of $R$ over the coefficient ring of $R$.
##### `One(F): FldFunRat -> FldFunRatElt`
##### `Identity(F): FldFunRat -> FldFunRatElt`
##### `Zero(F): FldFunRat -> FldFunRatElt`
##### `Representative(F): FldFunRat -> FldFunRatElt`
##### `Example: Creating Rational Function Field Elements (ex-64b44d)`
We create the field of rational functions over the integers in a single variable $w$.
```magma
> R := PolynomialRing(Integers());
> F := FieldOfFractions(R);
> F ! x+3;
w + 3
> F ! [ x, x-1 ];
w/(w - 1)
```
### Structure Operations
#### Related Structures
##### `IntegerRing(F): FldFunRat -> RngPol`
##### `RingOfIntegers(F): FldFunRat -> RngPol`
Given the rational function field $F$ this returns the polynomial ring from which $F$ was constructed as its field of fractions.
##### `BaseRing(F): FldFunRat -> Rng`
##### `CoefficientRing(F): FldFunRat -> Rng`
The coefficient ring of the (ring of integers of) the rational function field $F$.
##### `Rank(F): FldFunRat -> RngIntElt`
The rank (number of indeterminates) of the rational function field $F$.
##### `ValuationRing(F): FldFunRat -> RngVal`
Given the rational function field $F$ for which the coefficients come from a field, this returns the valuation ring of $F$ with respect to the valuation given by the degree. This valuation ring consists of those rational functions $g/h$ for which the degree of $h$ is greater than or equal to that of $g$.
##### `ValuationRing(F, f): FldFunRat, RngUPolElt -> RngVal`
Given the rational function field $F$ for which the coefficients come from a field, and an irreducible polynomial $f$ in the ring of integers of $F$, this returns the valuation ring of $F$ with respect to the valuation associated with $f$. This valuation ring consists of those rational functions $g/h$ for which $f$ divides $g$ but not $h$.
##### `Category(R): FldFunRat -> Cat`
##### `Parent(R): FldFunRat -> Pow`
##### `PrimeRing(R): FldFunRat -> Rng`
#### Invariants
##### `Characteristic(F): FldFunRat -> FldFunRatElt`
#### Ring Predicates and Booleans
##### `IsCommutative(F): FldFunRat -> BoolElt`
##### `IsUnitary(F): FldFunRat -> BoolElt`
##### `IsFinite(F): FldFunRat -> BoolElt`
##### `IsOrdered(F): FldFunRat -> BoolElt`
##### `IsField(F): FldFunRat -> BoolElt`
##### `IsEuclideanDomain(F): FldFunRat -> BoolElt`
##### `IsPID(F): FldFunRat -> BoolElt`
##### `IsUFD(F): FldFunRat -> BoolElt`
##### `IsDivisionRing(F): FldFunRat -> BoolElt`
##### `IsEuclideanRing(F): FldFunRat -> BoolElt`
##### `IsPrincipalIdealRing(F): FldFunRat -> BoolElt`
##### `IsDomain(F): FldFunRat -> BoolElt`
##### `F eq G: FldFunRat, Rng -> BoolElt`
##### `F ne G: FldFunRat, Rng -> BoolElt`
#### Homomorphisms
In its general form a ring homomorphism taking a function field $R(x_1, \ldots, x_n)$ as domain requires $n+1$ pieces of information, namely, a map (homomorphism) telling how to map the coefficient ring $R$ together with the images of the $n$ indeterminates.
##### `hom< P -> S | f, y₁, ..., yₙ >: FldFunRat, Rng -> Map`
##### `hom< P -> S | y₁, ..., yₙ >: FldFunRat, Rng -> Map`
Given a function field $F=R(x_1,\ldots, x_n)$, a ring $S$, a map $f : F\ar S$, and $n$ elements $y_1, \ldots, y_n\in S$, create the homomorphism $g : F\ar S$ by applying the rules of $g(rx_1^{a_1}\cdots x_n^{a_n})=f(r)y_1^{a_1}\cdots y_n^{a_n}$ for monomials, linearity for polynomials, i.e., $g(M+N)=g(M)+g(N)$, and division for fractions, i.e., $g(n/d)=g(n)/g(d)$. The coefficient ring map may be omitted, in which case the coefficients are mapped into $S$ by the unitary homomorphism sending $1_R$ to $1_S$. Also, the images $y_i$ are allowed to be from a structure that allows automatic coercion into $S$.
##### `Example: Homomorphism to a Number Field (ex-bca4bf)`
In this example we map ${\mathbb{Q}}(x, y)$ into the number field ${\mathbb{Q}}(\root 3 \of 2, \sqrt{5})$ by sending $x$ to $\root 3 \of 2$ and $y$ to $\sqrt{5}$ and the identity map on the coefficients (which we omit).
```magma
> Q := RationalField();
> F := FunctionField(Q, 2);
> A := PolynomialRing(IntegerRing());
> N := NumberField([a^3-2, a^2+5]);
> h := hom< F -> N | z, w >;
> h(x^11*y^3-x+4/5*y-13/4);
-40*w*z^2 - z + 4/5*w - 13/4
> h(x/3);
1/3*z
> h(1/x);
1/2*z^2
> 1/z;
1/2*z^2
```
### Element Operations
#### Arithmetic
##### `+ a: FldFunRatElt -> FldFunRatElt`
##### `- a: FldFunRatElt -> FldFunRatElt`
##### `a + b: FldFunRatElt, FldFunRatElt -> FldFunRatElt`
##### `a - b: FldFunRatElt, FldFunRatElt -> FldFunRatElt`
##### `a * b: FldFunRatElt, FldFunRatElt -> FldFunRatElt`
##### `a / b: FldFunRatElt, FldFunRatElt -> FldFunRatElt`
##### `a ^ k: FldFunRatElt, RngIntElt -> FldFunRatElt`
#### Equality and Membership
##### `a eq b: FldFunRatElt, FldFunRatElt -> BoolElt`
##### `a ne b: FldFunRatElt, FldFunRatElt -> BoolElt`
##### `a in F: FldFunRatElt, FldFunRat -> BoolElt`
##### `a notin F: FldFunRatElt, FldFunRat -> BoolElt`
#### Numerator, Denominator, and Degree
##### `Numerator(f): FldFunRatElt -> RngElt`
##### `Numerator(f): RngUPolElt -> RngUPolElt`
##### `Numerator(f): RngMPolElt -> RngMPolElt`
Given a rational function $f\in K$, the field of fractions of $R$, return the numerator $P$ of $f=P/Q$ as an element of the polynomial ring $R$.
##### `Denominator(f): FldFunRatElt -> RngElt`
##### `Denominator(f): RngUPolElt -> RngUPolElt`
##### `Denominator(f): RngMPolElt -> RngMPolElt`
Given a rational function $f\in K$, the field of fractions of $R$, return the denominator $Q$ of $f=P/Q$ as an element of the polynomial ring $R$.
##### `Degree(f): FldFunRatElt -> RngIntElt`
Given a rational function $f$ in a univariate function field, return the degree of $f$ as an integer (the maximum of the degree of the numerator of $f$ and the degree of the denominator of $f$).
##### `TotalDegree(f): FldFunRatElt -> RngIntElt`
Given a rational function $f$ in a multivariate function field, return the total degree of $f$ as an integer (the total degree of the numerator of $f$ minus the total degree of the denominator of $f$).
##### `WeightedDegree(f): FldFunRatElt -> RngIntElt`
Given a rational function $f$ in a multivariate function field, return the weighted degree of $f$ as an integer (the weighted degree of the numerator of $f$ minus the weighted degree of the denominator of $f$).
##### `Numerator(f, R): FldFunRatElt -> RngElt`
##### `Denominator(f, R): FldFunRatElt -> RngElt`
Return the numerator or denominator of $f$ with respect to the ring of integers $R$ of the rational function field $F$ containing $f$. The ring $R$ may be a polynomial ring or a valuation ring.
#### Predicates on Ring Elements
##### `IsZero(a): FldFunRatElt -> BoolElt`
##### `IsOne(a): FldFunRatElt -> BoolElt`
##### `IsMinusOne(a): FldFunRatElt -> BoolElt`
##### `IsNilpotent(a): FldFunRatElt -> BoolElt`
##### `IsIdempotent(a): FldFunRatElt -> BoolElt`
##### `IsUnit(a): FldFunRatElt -> BoolElt`
##### `IsZeroDivisor(a): FldFunRatElt -> BoolElt`
##### `IsRegular(a): FldFunRatElt -> BoolElt`
#### Evaluation
##### `Evaluate(f, r): FldFunRatUElt, RngElt -> FldFunRatUElt`
Given a univariate rational function $f$ in $F$, return the rational function in $F$ obtained by evaluating the indeterminate in $r$, which must be from (or coercible into) the coefficient ring of the integers of $F$.
##### `Evaluate(f, v, r): FldFunRatMElt, RngIntElt, RngElt -> FldFunRatMElt`
Given a multivariate rational function $f$ in $F$, return the rational function in $F$ obtained by evaluating the $v$-th variable in $r$, which must be from (or coercible into) the coefficient ring of the integers of $F$.
##### `Evaluate(f, S): FldFunRatMElt, [RngElt] -> RngElt`
Given a multivariate rational function $f$ in $F$ return the result of evaluating the $v$-th variable at the $v$-th element of the sequence $S$ which should be a sequence of elements coercible into the coefficient ring of the integers of $F$.
#### Decomposition
The following function provides an inverse operation to composing rational functions.
##### `Decomposition(f): FldFunRatUElt -> [[FldFunRatUElt]]`
```magma
All: BoolElt Default: true
```
Given a rational function $f$ defined over a field $k$, return all complete decompositions $[f_1,\ldots, f_r]$ of $f$ such that $f(t) = f_r(\ldots(f_1(t))$. These decompositions are found, as explained in [[ACvHS17](https://docs.magma-maths.org/references.md#cite-szutkoski)], by computing the subfield lattice of $k(t)/k(f)$, which in turn, is found by first computing the principal subfields of this extension and then computing all intersections of principal subfields. These intersections are computed by joining partitions.
The routines in this package are those developed by Peter Fleischmann and Jonas Szutkoski.
##### `Example: Decomposition of a Rational Function (ex-221d21)`
We show a decomposition of a rational function and the corresponding evaluations which retrieve the function which was decomposed.
```magma
> k:=GF(3);
> F:=FunctionField(k);
> f:=(t^10 + 2*t^8 + 2*t^4 + t^2)/(t^12 + 2*t^6 + 1);
> Decomposition(f);
[
[
t^2,
t/(t^2 + 1),
(2*t^3 + t^2 + 2*t)/(t^3 + 2)
],
[
t/(t^2 + 2),
t^2,
t/(t^3 + 1)
],
[
t/(t^2 + 1),
t^3 + 2*t,
t^2
],
[
t/(t^2 + 1),
t^2 + t,
t^3 + t^2
],
[
t/(t^2 + 1),
t^2 + 2*t,
t^3 + t^2
],
[
t/(t^2 + 1),
t^2,
t^3 + t^2 + t
]
]
> Decomposition(f : All := false);
[
[
t^2,
t/(t^2 + 1),
(2*t^3 + t^2 + 2*t)/(t^3 + 2)
]
]
> Evaluate($1[1][3], Evaluate($1[1][2], $1[1][1])) eq f;
true
```
#### Derivative
##### `Derivative(f): FldFunRatUElt -> FldFunRatUElt`
Given a univariate rational function $f$, return the first derivative of $f$ with respect to its variable.
##### `Derivative(f, k): FldFunRatUElt, RngIntElt -> FldFunRatUElt`
Given a univariate rational function $f$, return the $k$-th derivative of $f$ with respect to its variable. $k$ must be non-negative.
##### `Derivative(f, v): FldFunRatMElt, RngIntElt -> FldFunRatMElt`
Given a multivariate rational function $f$, return the first derivative of $f$ with respect to variable number $v$.
##### `Derivative(f, v, k): FldFunRatMElt, RngIntElt, RngIntElt -> FldFunRatMElt`
Given a multivariate rational function $f$, return the $k$-th derivative of $f$ with respect to variable number $v$. $k$ must be non-negative.
#### Partial Fraction Decomposition
##### `PartialFractionDecomposition(f): FldFunRatUElt -> [ ]`
Given a univariate rational function $f$ in $F=K(x)$, return the (unique) complete partial fraction decomposition of $f$. The decomposition is returned as a (sorted) sequence $Q$ consisting of triples, each of which is of the form $$ where $d$ is the denominator, $k$ is the multiplicity of the denominator, and $n$ is the corresponding numerator, and also $d$ is irreducible and the degree of $n$ is strictly less than the degree of $d$. Thus $f$ equals the sum of the $n_t/{d_t}^{k_t}$, where $t$ ranges over the triples contained in $Q$. If $f$ is improper (the degree of its numerator is greater than or equal to the degree of its denominator), then the first triple of $Q$ will be of the form $<1, 1, q>$ where $q$ is the quotient of the numerator of $f$ by the denominator of $f$.
##### `SquarefreePartialFractionDecomposition(f): FldFunRatUElt -> [ ]`
Given a univariate rational function $f$ in $F=K(x)$, return the (unique) complete squarefree partial fraction decomposition of $f$. The decomposition is returned as a (sorted) sequence $Q$ consisting of triples, each of which is of the form $$ where $d$ is the denominator, $k$ is the multiplicity of the denominator, and $n$ is the corresponding numerator, and also $d$ is squarefree and the degree of $n$ is strictly less than the degree of $d$. Thus $f$ equals the sum of the $n_t/{d_t}^{k_t}$, where $t$ ranges over the triples contained in $Q$. If $f$ is improper (the degree of its numerator is greater than or equal to the degree of its denominator), then the first triple of $Q$ will be of the form $<1, 1, q>$ where $q$ is the quotient of the numerator of $f$ by the denominator of $f$.
##### `Example: Partial Fraction Decomposition (ex-2ba727)`
We compute the squarefree and complete (irreducible) partial fraction decompositions of a fraction in $Q(t)$.
```magma
> F := FunctionField(RationalField());
> P := IntegerRing(F);
> f := ((t + 1)^8 - 1) / ((t^3 - 1)*(t + 1)^2*(t^2 - 4)^2);
> SD := SquarefreePartialFractionDecomposition(f);
> SD;
[
,
,
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in SD] eq f;
> D := PartialFractionDecomposition(f);
> D;
[
,
,
,
,
,
,
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in D] eq f;
true
```
Note that doing the same operation in the function field $Z(t)$ must modify the numerators and denominators to be integral but the result is otherwise the same.
```magma
> F := FunctionField(IntegerRing());
> P := IntegerRing(F);
> f := ((t + 1)^8 - 1) / ((t^3 - 1)*(t + 1)^2*(t^2 - 4)^2);
> D := PartialFractionDecomposition(f);
> D;
[
<2646*x - 5292, 1, -3683>,
<63*x - 126, 2, 25830>,
<36*x - 36, 1, 85>,
<108*x + 108, 1, 1>,
<18*x + 18, 2, 18>,
<18*x + 36, 1, 1>,
<147*x^2 + 147*x + 147, 1, -5*x - 8>
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in D] eq f;
true
```
Finally, we compute the partial fraction decomposition of a fraction in a function field whose coefficient ring is a multivariate function field.
```magma
> R := FunctionField(IntegerRing(), 2);
> F := FunctionField(R);
> P := IntegerRing(F);
> f := 1 / ((t^2 - a)^2*(t + b)^2*t^3);
> SD := SquarefreePartialFractionDecomposition(f);
> SD;
[
,
,
,
,
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in SD] eq f;
true
> D := PartialFractionDecomposition(f);
> D;
[
,
,
,
,
,
,
]
> // Check appropriate sum equals f:
> &+[F!t[3] / F!t[1]^t[2]: t in D] eq f;
true
```
### Padé-Hermite Approximants
#### Introduction
A given rational function $F(z)\in k(z)$ over a field $k$ can be written as a power series $f(z)$ in the completion $k((z))$ of $k(z)$ at the place $(z)$. It thus is a good approximation of $f(z)$ in the sense that $F(z)$ is equal to $f(z)$ up to infinite order.
Padé-Hermite approximants deal with the converse. Given a (formal) power series $f(z)\in k[[z]]$ and some non- negative integers $n_P$ and $n_Q$, find polynomials $P(z), Q(z)\in k[z]$ of degrees at most $n_P$ and $n_Q$, respectively, such that $P(z)*f(z)-Q(z) = O(z^{n_P+n_Q+1})$ holds. In other words, $P/Q(z)$ is an approximation for $f(z)$ for polynomials $P$ and $Q$ of limited degree in $z$. In this notation the pair $[P,Q]$ is known as the Padé-Hermite approximant for the power series tuple $(f,-1)$.
The idea of approximating one power series by two polynomials can be extended to approximating several power series at the same time as follows. Consider the vector $\underline{f}^T:= (f_1, f_2, \ldots, f_m)^T$ in the $m$-dimensional vector space $k((z))^m$ over a power series ring. Let $\underline{n}=(n_1, n_2,\ldots, n_m)$ be an m-tuple of non-negative integers. A *Padé-Hermite approximant* of $\underline{f}^T(z)$ of type $\underline{n}$ is a non-zero vector of polynomials $\underline{P}=(P_1,P_2,\ldots,P_m)$ in $k[z]^m$ such that $\underline{P}\cdot\underline{f}^T = O(z^{N})$ , $N=n_1+n_2+\cdots+n_m+m-1$, is satisfied. A non-trivial Padé-Hermite approximant always exists. The approximant is contained in the sub-space $V_{\underline{f},N}=\{ \underline{Q}\in k[z]^m : \underline{Q}\cdot\underline{f}^T = O(z^{N})\}$.
The implementation of Padé-Hermite approximants is based on [[Der94](https://docs.magma-maths.org/references.md#cite-derksen94)] and [[BL94a](https://docs.magma-maths.org/references.md#cite-bl94)]. They are implemented in Magma as sequences rather than vectors.
#### Ordering of Sequences
A Padé-Hermite approximant to some sequence of power series does not have to be unique. They can be ordered according to their maximum degree and their type of sequence. A sequence $\underline{P}=[P_1, P_2,..., P_m]$ is of degree $i$ if the maximum of the weak degrees of $P_1, P_2, \ldots, P_m$ is $i$. An extension of the definition maximum degree is when a distortion of non-negative integers $\underline{d}=[d_1, d_2, \ldots, d_m]$ on the degrees is allowed. In this case the maximum degree is defined as the maximum of $\deg{P_i}-d_i$. The type of $\underline{P}$ identifies the last $P_i$ in $\underline{P}$ whose weak degree equals the maximum degree of $\underline{P}$. Again a distortion on the degrees is allowed.
##### `MaximumDegree(f): SeqEnum -> RngIntElt`
```magma
Distortion: SeqEnum Default: []
```
MaximumDegree returns the degree of a sequence of polynomials or power series, defined as the maximum of the degrees of $f[i]-d[i]$, where $d$ is the distortion. The value -infinity is returned in the case that $f$ is weakly equal to the zero-sequence.
##### `Example: Maximum Degree of a Sequence (ex-ea74e8)`
```magma
> S := PowerSeriesRing(Rationals());
> f := [u+u^2, 2+u^2+u^3,0];
> MaximumDegree(f);
3
> MaximumDegree(f:Distortion:=[]);
3
> MaximumDegree(f:Distortion:=[0,2,1]);
2
> MaximumDegree([S|0,0]);
-Infinity
> MaximumDegree([O(u)]);
-Infinity
```
##### `TypeOfSequence(f): SeqEnum -> RngIntElt, RngIntElt`
```magma
Distortion: SeqEnum Default: []
```
Returns the highest index $i$ for those $f[i]$ whose (distorted) degree is weakly equal to the maximum of the degrees of all entries. The second integer returned is the maximum degree of the sequence.
##### `Example: Type of a Sequence (ex-8d75ca)`
```magma
> S := PowerSeriesRing(Rationals());
> f := [u+u^2, 2+u^2+u^3,0];
> TypeOfSequence(f);
2 3
> TypeOfSequence(f:Distortion:=[]);
2 3
> TypeOfSequence(f:Distortion:=[0,2,1]);
1 2
> TypeOfSequence([S|0,0]);
2 -Infinity
```
The Padé-Hermite approximant of $\underline{f}$ can be seen as an element of the free module $k[z]^m$ of rank $m$, or as an element of the sub-space $V_{\underline{f},N}$.
A free sub-model $V\subset k[z]^m$ is generated by $m$ polynomial vectors $\underline{Q}_i$, $1=1,2,\ldots,m$, such that $\underline{Q}_1(z), \underline{Q}_2(z), \ldots, \underline{Q}_m(z)$ form a minimal vector sequence of $V$. Such a sequence is defined as a sequence $S$ of $m$ vectors in $V$, such that $S[i]$ is a non-trivial polynomial vector in $V$ of minimal degree of type $i$, for $i=1,2,\ldots,m$. A minimal vector sequence is not unique.
Two variations on a minimal vector sequence are implemented. The first allows a distortion as an attribute. The sequence is the based on the distorted maximal degree. The second attribute sets the positive power $p$ of $z$ in each of the $\underline{Q}_i$ as follows. Instead of considering $\underline{Q}_i(z)$ for each $i$, one considers $\underline{Q}_i(z^p)$.
##### `MinimalVectorSequence(f, n): SeqEnum, RngIntElt -> SeqEnum`
```magma
Distortion: SeqEnum Default: []
Power : RngIntElt Default: 1
```
A minimal sequence of vectors $\underline{Q}_1, \underline{Q}_2, \ldots, \underline{Q}_m$ with respect to the sequence $f$ of length $m$ whose entries are polynomials or power series. The order of $\underline{Q}_i\cdot f$ is at least $m$.
##### `Example: Minimal Vector Sequence of Power Series (ex-67b928)`
```magma
> S := PowerSeriesRing(Rationals());
> f := [u+u^2, 2+u^2+u^3];
> seq := MinimalVectorSequence(f, 2);
> seq;
[
(u 0),
( -1 1/2*u)
]
> sums := [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
> sums;
[
u^2 + u^3,
-u^2 + 1/2*u^3 + 1/2*u^4
]
> seq := MinimalVectorSequence(f,3);
> #seq eq 2, seq[1], seq[2];
true (u^2 0)
(-1 + u 1/2*u)
```
##### `Example: Minimal Vector Sequence of Polynomials (ex-d1c2b6)`
```magma
> L := PolynomialRing(Rationals());
> f := [1+x, 3-x^2, 5+x+x^3-x^5];
> seq := MinimalVectorSequence(f, 10);
> seq[1];
(-2*x^2 + 6 -2*x - 2 0)
> sums := [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
> sums;
[
0,
0,
x^10
]
```
##### `Example: Minimal Vector Sequence of Higher Order (ex-328015)`
```magma
> S := PowerSeriesRing(Rationals());
> f := [2*u^4,2+u^3+u^6];
> seq := MinimalVectorSequence(f, 10);
> seq;
[
(1/2*u^6 0),
(-1/2 - 1/4*u^3 1/2*u^4)
]
> sums := [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
> sums;
[
u^10,
1/2*u^10
]
```
##### `Example: Minimal Vector Sequence with Distortion (ex-1d53c0)`
```magma
> S := PowerSeriesRing(Rationals());
> f := [1+u-7*u^2, 6-3*u+1/2*u^2-u^3, 5-u+u^2];
> seq := MinimalVectorSequence(f, 5);
> [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
[
0,
u^5,
0
]
> seq := MinimalVectorSequence(f, 5:Distortion:=[2,0,1]);
> [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
[
-7/2*u^5,
u^5,
0
]
> p:=2;
> seq := MinimalVectorSequence(f, 5:Distortion:=[2,0,1], Power:=p);
> sums := [&+([Q[i]*f[i]: i in [1..#f]]) : Q in seq];
> sums;
> mp:= mapS| x :-> (IsWeaklyZero(x) select 0
> else &+([Coefficient(x,i)*(S.1)^(p*i) : i in Exponents(x)]))
> + (ISA(Type(v),RngIntElt) select O((S.1)^(p*v))
> else S!0 where v := AbsolutePrecision(x))>;
> sums := [&+([mp(Q[i])*f[i]: i in [1..#f]]) : Q in seq];
> sums;
[
u^6 + u^7 - 7*u^8,
u^5 - 23/3*u^6,
-5/3*u^5 + 35/3*u^6
]
```
#### Approximants
The Padé-Hermite approximant of type $\underline{d}=[d_1,d_2,\ldots,d_m]$ with respect to the tuple $\underline{f}^T\in k((z))^m$ is an element of the space $V_{\underline{f},N}=\{ \underline{Q}\in k[z]^m : \underline{Q}\cdot\underline{f}^T = O(z^{N})\}$ for $N$ equal to $d_1+d_2+\cdots+d_m+m-1$. This space is generated by the vectors in the minimal vector sequence with respect to $\underline{f}$ with distortion $\underline{d}$. The routine [`PadeHermiteApproximant`](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/padehermite.md#function-padhermapp) returns one that is smallest with respect to the degree on sequences. The input sequence $\underline{f}$ must be a sequence of polynomial ring elements, or be a power series sequence. While the Padé Hermite approximants theoretically are polynomials, Magma returns them as elements of the same ring the entries of $\underline{f}$ are contained in.
##### `PadeHermiteApproximant(f, d): SeqEnum, SeqEnum -> ModTupRngElt, SeqEnum, RngIntElt`
```magma
Power: RngIntElt Default: 1
```
Returns a Padé-Hermite form $\underline{P}$ of $f$ with distortion $d$, smallest with respect to the degree on sequences, and the corresponding minimal vector sequence. The third argument returned is the order in the order term of $\underline{P}\cdot f$.
##### `Example: Pade-Hermite Approximants (ex-05a6f0)`
This example can be found on page 813 in [[BL94a](https://docs.magma-maths.org/references.md#cite-bl94)].
```magma
> S := PowerSeriesRing(Rationals());
> f := [1,u,u/(1-u^4)+u^10+O(u^16),u/(1+u^4)+u^12+O(u^16)];
> pade, padebasis, ord := PadeHermiteApproximant(f,[2,2,2,2]);
> pade, ord;
( u -1 0 0)
11
> BaseRing(Parent(pade)) eq S;
true
> MinimalVectorSequence(f,10);
[
( u -1 0 0),
( 0 u^4 -1/2 1/2),
( 0 1 - u^4 -1/2 + u^4 -1/2),
( 0 -u 1/2*u 1/2*u)
]
>
> p := 2;
> seq := MinimalVectorSequence(f,10: Distortion :=[2,2,2,2],Power := p);
> seq;
[
(u^5 0 0 0),
( 0 u^2 -1/2 1/2),
( 0 1 - u^2 -1/2 + u^2 -1/2),
( 0 -u 1/2*u 1/2*u)
]
> mp:= mapS| x :-> (IsWeaklyZero(x) select 0
> else &+([Coefficient(x,i)*(S.1)^(p*i) : i in Exponents(x)]))
> + (ISA(Type(v),RngIntElt) select O((S.1)^(p*v))
> else S!0 where v := AbsolutePrecision(x))>;
> sums := [&+([mp(Q[i])*f[i]: i in [1..#f]]) : Q in seq];
> [Valuation(v) : v in sums];
[ 10, 10, 10, 11 ]
> sums;
[
u^10,
-1/2*u^10 + 1/2*u^12 - u^13 + O(u^16),
-1/2*u^10 - 1/2*u^12 + u^13 + u^14 + O(u^16),
u^11 + 1/2*u^12 + 1/2*u^14 + O(u^18)
]
```
##### `Example: Pade-Hermite Approximant with Distortion (ex-e98d49)`
This example covers the example on page 815 in [[BL94a](https://docs.magma-maths.org/references.md#cite-bl94)].
```magma
> S := PowerSeriesRing(Rationals());
> f := [1,u, -1-u^4-2*u^8+u^10+u^11-u^12+O(u^16),-u-u^5-u^9-u^14-u^15+O(u^16)];
> dist:=[2,2,3,3];
> seq := MinimalVectorSequence(f,13:Distortion:=dist);
> pade, padebasis, ord := PadeHermiteApproximant(f,dist);
> pade, ord;
(-u 1 0 0)
13
> padebasis;
[
(-u 1 0 0),
(1/2*u -1/2*u^4 1/2*u + 1/2*u^3 + 1/2*u^4 -1/2*u^2 - 1/2*u^3 - u^4),
(-1/2 1/2*u^3 -1/2 - 1/2*u^2 - 1/2*u^3 - u^4 1/2*u + 1/2*u^2 + 2*u^3),
( -u 0 0 -1 + u^4)
]
> padebasis eq seq;
true
> [[Valuation(w[i]): i in [1..Degree(w)]] : w in seq];
[
[ 1, 0, Infinity, Infinity ],
[ 1, 4, 1, 2 ],
[ 0, 3, 0, 1 ],
[ 1, Infinity, Infinity, 0 ]
]
> [[MaximumDegree([w[i]])-dist[i]: i in [1..Degree(w)] ] : w in seq];
[
[ -1, -2, -Infinity, -Infinity ],
[ -1, 2, 1, 1 ],
[ -2, 1, 1, 0 ],
[ -1, -Infinity, -Infinity, 1 ]
]
> p:=2;
> seq := MinimalVectorSequence(f,12:Distortion:=dist,Power:=p);
> seq;
[
( u - u^3 0 u - 2*u^3 0),
(-1 - u + u^2 -u^3 -1 - u + 2*u^2 + u^3 -u^3),
(u + u^2 - u^3 0 u + u^2 - 2*u^3 - u^4 0),
( 0 1 0 1 - u^2)
]
> seq[1]-seq[3];
( -u^2 0 -u^2 + u^4 0)
> mp:= mapS| x :-> (IsWeaklyZero(x) select 0
> else &+([Coefficient(x,i)*(S.1)^(p*i) : i in Exponents(x)]))
> + (ISA(Type(v),RngIntElt) select O((S.1)^(p*v))
> else S!0 where v := AbsolutePrecision(x))>;
> [Valuation(&+([mp(Q[i])*f[i]: i in [1..#f]])) : Q in seq];
[ 12, 12, 13, 13 ]
```
##### `Example: Degrees and Valuations of Approximants (ex-ea09f6)`
This example considers the example on page 816 in [[BL94a](https://docs.magma-maths.org/references.md#cite-bl94)].
```magma
> S := PowerSeriesRing(Rationals());
> f := [1,u,-1-u^4-2*u^8+u^10+O(u^12),-u-u^5-u^9+u^10+O(u^12)];
> dist := [2,2,3,3];
> seq := MinimalVectorSequence(f,12: Distortion := dist);
> [[MaximumDegree([w[i]])-dist[i]: i in [1..Degree(w)] ] : w in seq];
[
[ -1, -2, -Infinity, -Infinity ],
[ -2, 1, 0, 0 ],
[ -Infinity, -Infinity, 1, 0 ],
[ -1, -Infinity, 0, 1 ]
]
> [Valuation(&+([(Q[i])*f[i]: i in [1..#f]]) ) : Q in seq];
[ Infinity, 12, 12, 12 ]
> [MaximumDegree([ &+([(Q[i])*f[i]: i in [1..#f]]) ]) : Q in seq];
[ -Infinity, -Infinity, 14, -Infinity ]
> PadeHermiteApproximant(f,[2,2,3,3]);
> p := 2;
> seq := MinimalVectorSequence(f,12:Distortion:=[2,2,3,3],Power:=p);
> seq;
[
( 1 - u^2 -1 1 - 2*u^2 -1 + u^2),
( 0 -u^4 0 -u^4),
( -u -1 -u + u^3 -1 + u^2),
( 0 u 0 u - u^3)
]
> [[MaximumDegree([w[i]])-dist[i]: i in [1..Degree(w)] ] : w in seq];
[
[ 0, -2, -1, -1 ],
[ -Infinity, 2, -Infinity, 1 ],
[ -1, -2, 0, -1 ],
[ -Infinity, -1, -Infinity, 0 ]
]
> mp:= mapS| x :-> (IsWeaklyZero(x) select 0
> else &+([Coefficient(x,i)*(S.1)^(p*i) : i in Exponents(x)]))
> + (ISA(Type(v),RngIntElt) select O((S.1)^(p*v))
> else S!0 where v := AbsolutePrecision(x))>;
>
> [Valuation(&+([mp(Q[i])*f[i]: i in [1..#f]])) : Q in seq];
[ 12, 13, 12, 12 ]
```
A variant of the Padé-Hermite approximant is when the exponent in the order term is set rather that the type of the sequence. It is also possible to let $\underline{f}$ be a sequence such that its entries themselves are vectors of polynomials or power series.
##### `PadeHermiteApproximant(f, m): SeqEnum, RngIntElt -> ModTupRngElt, SeqEnum`
```magma
Power: RngIntElt Default: 1
```
Returns a Padé-Hermite form of minimal degree in the corresponding minimal vector sequence, such that its inproduct with $f$ has order at least $m$. The second argument returned is the corresponding minimal vector sequence.
##### `Example: Pade-Hermite Approximants of Vectors (ex-b5d830)`
This example can be found on page 813 in [[BL94a](https://docs.magma-maths.org/references.md#cite-bl94)].
```magma
> S := PowerSeriesRing(Rationals());
> f := [Vector([1]), Vector([u])];
> pade, seq := PadeHermiteApproximant(f,3);
Calculating the Pade'-Hermite approximant for the sequence [
1,
u
]
with order term 3 and power 1 .
> pade;
( u -1)
> seq;
[
( u -1),
( 0 u^2)
]
> mat := Matrix([Eltseq(v): v in f]);
> pade*mat;
(0)
> PadeHermiteApproximant([1,u],5);
( u -1)
[
( u -1),
( 0 u^4)
]
> PadeHermiteApproximant(f,3:Power:=2);
> g:= [Vector([1,0,0]), Vector([0,1,0]), Vector([1+u,2+u^2,u^3])];
> pade := PadeHermiteApproximant(g,5);
Calculating the Pade'-Hermite approximant for the sequence [
1,
u,
1 + 2*u + u^3 + u^7 + u^11
]
with order term 15 and power 3 .
> pade;
(-u^3 - u^4 -2*u^3 u^3)
> pade*Matrix([Eltseq(v): v in g]);
( 0 u^5 u^6)
```
##### `Example: Approximants of Sine, Cosine, and Exponential (ex-15b388)`
This example considers Padé-Hermite approximants for some series that have non-trivial power series expansions.
```magma
> S := PowerSeriesRing(Rationals());
> f := [Sin(u), Cos(u), Exp(u)];
> [Valuation(f[i]) : i in [1..#f]], [Degree(f[i]) : i in [1..#f]];
[ 1, 0, 0 ]
[ 19, 20, 20 ]
> [AbsolutePrecision(f[i]) : i in [1..#f]];
[ 21, 22, 21 ]
> dist := [3,2,5];
> pade, seq, ord := PadeHermiteApproximant(f,dist);
> 1/420*pade;
(-1275 - 255*u + 45*u^2 + 5*u^3 120 + 495*u + 75*u^2 -120 + 900*u - 600*u^2 +
160*u^3 - 20*u^4 + u^5)
> ord eq &+(dist)+#f-1, ord;
true 12
> [Degree(pade[i]) : i in [1..Degree(pade)]];
[ 3, 2, 5 ]
> g:= [Cos(2*u)*(u+1)+3,Cos(u)^2+u*Cos(u)+1,Cos(2*u)+1,Cos(u)];
> pade, basis := PadeHermiteApproximant(g,20);
> 131/75880*pade;
( 2 -4 + 2*u -3*u 4*u - 2*u^2)
> h := [ 1+u^2-u^7+u^12, Sin(u), Exp(u) ];
> dist:=[3,1,2];
> seq := MinimalVectorSequence(h,8:Distortion := dist);
> sums := [&+([Q[i]*h[i]: i in [1..#f]]) : Q in seq];
> [Valuation(s) : s in sums];
[ 8, 8, 8 ]
> [[MaximumDegree([w[i]]): i in [1..Degree(w)] ] : w in seq];
[
[ 4, 1, 2 ],
[ 4, 2, 2 ],
[ 3, 1, 2 ]
]
> [[MaximumDegree([w[i]])-dist[i]: i in [1..Degree(w)] ] : w in seq];
[
[ 1, 0, 0 ],
[ 1, 1, 0 ],
[ 0, 0, 0 ]
]
```
## Algebraic Function Fields
- [Introduction](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/introduction.md)
- [Representations of Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/introduction.md#representations-of-fields)
- [Creation of Algebraic Function Fields and their Orders](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md)
- [Creation of Algebraic Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#creation-of-algebraic-function-fields)
- [`ext< K | f >: FldFunRat, RngUPolElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#constructor-constructor-ext-fldfunrat-rngupolelt-fldfun)
- [`ext< K | f >: FldFun, RngUPolElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#constructor-constructor-ext-fldfun-rngupolelt-fldfun)
- [`FunctionField(f : parameters): RngUPolElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-functionfield-rngupolelt)
- [`FunctionField(f : parameters): RngMPolElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-functionfield-rngmpolelt)
- [`FunctionField(S): [RngUPolElt] → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-functionfield-rngupolelt-2)
- [`HermitianFunctionField(p, d): RngIntElt, RngIntElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-hermitianfunctionfield-rngintelt-rngintelt)
- [`HermitianFunctionField(q): RngIntElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-hermitianfunctionfield-rngintelt)
- [`sub: FldFun, [] → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#constructor-constructor-sub-fldfun-fldfun)
- [`sub: FldFun, [] → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#constructor-constructor-sub-fldfun-fldfun-2)
- [`AssignNames(~F, s): FldFun, [ MonStgElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#procedure-assignnames-fldfun-monstgelt-ref)
- [`AssignNames(~a, s): FldFunElt, [ MonStgElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#procedure-assignnames-fldfunelt-monstgelt-ref)
- [`FunctionField(R): Rng → FldFunG`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-functionfield-rng)
- [Example: Creating a Function Field Extension](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#example-ex-4f7d5b)
- [Example: Creating a Relative Extension](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#example-ex-b33520)
- [Example: Creating a Non-Simple Extension](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#example-ex-5f689c)
- [Example: Hermitian Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#example-ex-5c6691)
- [Construction of Orders of Algebraic Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#construction-of-orders-of-algebraic-function-fields)
- [`EquationOrderFinite(F): FldFun → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-equationorderfinite-fldfun)
- [`MaximalOrderFinite(F): FldFun → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-maximalorderfinite-fldfun)
- [`EquationOrderInfinite(F): FldFun → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-equationorderinfinite-fldfun)
- [`MaximalOrderInfinite(F): FldFun → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-maximalorderinfinite-fldfun)
- [`IntegralClosure(R, F): Rng, FldFun → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-integralclosure-rng-fldfun)
- [`EquationOrder(O): RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-equationorder-rngfunord)
- [`MaximalOrder(O): RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-fldfung-max-ord-al)
- [`SetOrderMaximal(O, b): RngFunOrd, BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#procedure-setordermaximal-rngfunord-boolelt)
- [`ext: RngFunOrd, RngUPolElt → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#constructor-constructor-ext-rngfunord-rngupolelt-rngfunord)
- [Example: Creating Orders of a Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#example-ex-68faea)
- [Example: Integral Closure in a Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#example-ex-4ba3aa)
- [`Order(O, T, d): RngFunOrd, AlgMatElt, RngElt → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-order-rngfunord-algmatelt-rngelt)
- [`Order(O, M): RngFunOrd, ModDed → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-order-rngfunord-modded)
- [`Order(O, S): RngFunOrd, [FldFunElt] → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-order-rngfunord-fldfunelt)
- [`Simplify(O): RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-simplify-rngfunord)
- [`O1 + O2: RngFunOrd, RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#operation-op-plus-rngfunord-rngfunord)
- [`O1 meet O2: RngFunOrd, RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#operation-op-meet-rngfunord-rngfunord)
- [`AsExtensionOf(O1, O2): RngFunOrd, RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-asextensionof-rngfunord-rngfunord)
- [Example: Further Order Constructions](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#example-ex-b548f7)
- [Orders and Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#orders-and-ideals)
- [`MultiplicatorRing(I): RngFunOrdIdl → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-multiplicatorring-rngfunordidl)
- [`pMaximalOrder(O, p): RngFunOrd, RngFunOrdIdl → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-pmaximalorder-rngfunord-rngfunordidl)
- [`pMaximalOrder(O, p): RngFunOrd, RngElt → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-pmaximalorder-rngfunord-rngelt)
- [`pRadical(O, p): RngFunOrd, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-pradical-rngfunord-rngfunordidl)
- [`pRadical(O, p): RngFunOrd, RngElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/creation.md#function-pradical-rngfunord-rngelt)
- [Related Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md)
- [Parent and Category](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#parent-and-category)
- [`Category(F): FldFun → Cat`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-category-fldfun)
- [`Category(O): RngFunOrd → Cat`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-category-rngfunord)
- [`Parent(F): FldFun → Pow`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-parent-fldfun)
- [`Parent(O): RngFunOrd → Pow`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-parent-rngfunord)
- [Other Related Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#other-related-structures)
- [`PrimeRing(F): FldFun → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-primering-fldfun)
- [`PrimeField(F): FldFun → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-primefield-fldfun)
- [`PrimeRing(O): RngFunOrd → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-primering-rngfunord)
- [`ConstantField(F): FldFunG → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-constantfield-fldfung)
- [`DefiningConstantField(F): FldFunG → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-definingconstantfield-fldfung)
- [`ExactConstantField(F): FldFunG → Rng, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-exactconstantfield-fldfung)
- [`BaseRing(F): FldFun → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-basering-fldfun)
- [`BaseField(F): FldFun → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-basefield-fldfun)
- [`CoefficientRing(F): FldFun → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-coefficientring-fldfun)
- [`CoefficientField(F): FldFun → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-coefficientfield-fldfun)
- [`ISABaseField(F,G): Fld, Fld → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-isabasefield-fld-fld)
- [`BaseRing(O): RngFunOrd → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-basering-rngfunord)
- [`CoefficientRing(O): RngFunOrd → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-coefficientring-rngfunord)
- [`BaseRing(FF): FldFunOrd → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-basering-fldfunord)
- [`BaseField(FF): FldFunOrd → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-basefield-fldfunord)
- [`CoefficientRing(FF): FldFunOrd → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-coefficientring-fldfunord)
- [`CoefficientField(FF): FldFunOrd → Rng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-coefficientfield-fldfunord)
- [`SubOrder(O): RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-suborder-rngfunord)
- [`FunctionField(O): RngFunOrd → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-functionfield-rngfunord)
- [`FieldOfFractions(O): RngFunOrd → FldFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-fieldoffractions-rngfunord)
- [`FieldOfFractions(FF): FldFunOrd → FldFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-fieldoffractions-fldfunord)
- [`FieldOfFractions(F): FldFun → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-fieldoffractions-fldfun)
- [`Order(FF): FldFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-order-fldfunord)
- [`RationalExtensionRepresentation(F): FldFunG → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-rationalextensionrepresentation-fldfung)
- [`AbsoluteOrder(O): RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-absoluteorder-rngfunord)
- [`AbsoluteFunctionField(F): FldFunG → FldFunG`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-absolutefunctionfield-fldfung)
- [`UnderlyingRing(F): FldFunG → FldFunG`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-underlyingring-fldfung)
- [`UnderlyingField(F): FldFunG → FldFunG`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-underlyingfield-fldfung)
- [`UnderlyingRing(F, R): FldFunG, Rng → FldFunG`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-underlyingring-fldfung-rng)
- [`UnderlyingField(F, R): FldFunG, Rng → FldFunG`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-underlyingfield-fldfung-rng)
- [`Embed(F, L, a): FldFun, FldFun, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#procedure-embed-fldfun-fldfun-fldfunelt)
- [`Embed(F, L, s): FldFun, FldFun, [FldFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#procedure-embed-fldfun-fldfun-fldfunelt-2)
- [`HasEmbedding(O1, O2): RngFunOrd, RngFunOrd → Bool`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-hasembedding-rngfunord-rngfunord)
- [`CanComputeEmbedding(O1, O2): RngFunOrd, RngFunOrd → Bool`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-cancomputeembedding-rngfunord-rngfunord)
- [`Places(F): FldFunG → PlcFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-places-fldfung)
- [`DivisorGroup(F): FldFun → DivFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-divisorgroup-fldfun)
- [`DifferentialSpace(F): FldFun → DiffFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-differentialspace-fldfun)
- [Example: Constant and Coefficient Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#example-ex-a0477f)
- [Example: Underlying Rings of a Relative Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#example-ex-efbad3)
- [`WeilRestriction(E, n): FldFun, RngIntElt → FldFun, UserProgram`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-weilrestriction-fldfun-rngintelt)
- [`ConstantFieldExtension(F, E): FldFun, Rng → FldFun, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-constantfieldextension-fldfun-rng)
- [Example: Extending to the Exact Constant Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#example-ex-01df8e)
- [`MonicModel(F): FldFun → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-monicmodel-fldfun)
- [`Reduce(O): RngFunOrd → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-reduce-rngfunord)
- [`Localization(O, p): RngFunOrd, RngFunOrdIdl → RngVal, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-related.md#function-localization-rngfunord-rngfunordidl)
- [General Structure Invariants](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md)
- [`Characteristic(F): FldFun → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-characteristic-fldfun)
- [`Characteristic(O): RngFunOrd → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-characteristic-rngfunord)
- [`IsPerfect(F): Fld → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-isperfect-fld)
- [`Degree(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-degree-fldfung)
- [`Degree(F, G): FldFun, FldFun → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-degree-fldfun-fldfun)
- [`Degree(O): RngFunOrd → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-degree-rngfunord)
- [`AbsoluteDegree(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-absolutedegree-fldfung)
- [`AbsoluteDegree(O): RngFunOrd → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-absolutedegree-rngfunord)
- [`DefiningPolynomial(F): FldFun → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-definingpolynomial-fldfun)
- [`DefiningPolynomial(O): RngFunOrd → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-definingpolynomial-rngfunord)
- [`DefiningPolynomials(F): FldFun → [RngUPolElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-definingpolynomials-fldfun)
- [`DefiningPolynomials(O): RngFunOrd → [RngUPolElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-definingpolynomials-rngfunord)
- [`Basis(F): FldFunG → SeqEnum[FldFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-basis-fldfung)
- [`Basis(O): RngFunOrd → SeqEnum[FldFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-basis-rngfunord)
- [`Basis(O, R): RngFunOrd, Rng → SeqEnum[RngElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-basis-rngfunord-rng)
- [`TransformationMatrix(O1, O2): RngFunOrd, RngFunOrd → AlgMatElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-transformationmatrix-rngfunord-rngfunord)
- [`CoefficientIdeals(O): RngFunOrd → [RngFunOrdIdl]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-coefficientideals-rngfunord)
- [`BasisMatrix(O): RngFunOrd → AlgMatElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-basismatrix-rngfunord)
- [`PrimitiveElement(O): RngFunOrd → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-primitiveelement-rngfunord)
- [`Discriminant(O): RngFunOrd → .`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-discriminant-rngfunord)
- [`AbsoluteDiscriminant(O): RngFunOrd → .`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-absolutediscriminant-rngfunord)
- [`DimensionOfExactConstantField(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-dimensionofexactconstantfield-fldfung)
- [`DegreeOfExactConstantField(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-degreeofexactconstantfield-fldfung)
- [`Genus(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-genus-fldfung)
- [Example: Invariants of a Relative Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#example-ex-a8d7ef)
- [Example: Invariants of a Non-Simple Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#example-ex-ec93f9)
- [`GapNumbers(F): FldFunG → SeqEnum[RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-gapnumbers-fldfung)
- [`GapNumbers(F, P): FldFunG, PlcFunElt → SeqEnum[RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-gapnumbers-fldfung-plcfunelt)
- [`GapNumbers(P): PlcFunElt → SeqEnum[RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-gapnumbers-plcfunelt)
- [`SeparatingElement(F): FldFunG → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-separatingelement-fldfung)
- [`RamificationDivisor(F): FldFunG → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-ramificationdivisor-fldfung)
- [`WeierstrassPlaces(F): FldFunG → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-weierstrassplaces-fldfung)
- [`WronskianOrders(F): FldFunG → [RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-wronskianorders-fldfung)
- [`Different(O): RngFunOrd → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-different-rngfunord)
- [`Index(O, S): RngFunOrd, RngFunOrd → Any`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-invar.md#function-index-rngfunord-rngfunord)
- [Galois Groups](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md)
- [`GaloisGroup(f): RngUPolElt → GrpPerm, [ RngElt ], GaloisData`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-galoisgroup-rngupolelt)
- [`GaloisGroup(F): FldFun → GrpPerm, [RngElt], GaloisData`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-galoisgroup-fldfun)
- [`GaloisGroup(F): FldFun[FldFun[FldFunRat]] → GrpPerm, [RngElt], GaloisData`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-galoisgroup-fldfun-fldfun-fldfunrat)
- [`GaloisGroup(F): FldFun[FldFin] → GrpPerm, [RngElt], GaloisData`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-galoisgroup-fldfun-fldfin)
- [`GaloisGroup(F): FldFun[FldFunRat] → GrpPerm, [RngElt], GaloisData`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-galoisgroup-fldfun-fldfunrat)
- [`GaloisGroup(F): FldFun[FldRat] → GrpPerm, [RngElt], GaloisData`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-galoisgroup-fldfun-fldrat)
- [Example: Galois Group over a Rational Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#example-ex-0264c2)
- [Example: Galois Groups in Positive Characteristic](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#example-ex-93f3e8)
- [`GeometricGaloisGroup(f): RngUPolElt → GrpPerm, RngUPolElt, GaloisData`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-geometricgaloisgroup-rngupolelt)
- [`HilbertIrreducibilityCurves(f): RngUPolElt → SetEnum, SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-hilbertirreducibilitycurves-rngupolelt)
- [Example: Geometric Galois Group](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#example-ex-2194f2)
- [Splitting Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#splitting-fields)
- [`GaloisSplittingField(f): RngUPolElt → FldFun, [FldFunElt], GrpPerm, [[FldFunElt]]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-galoissplittingfield-rngupolelt)
- [Example: Galois Splitting Field of a Polynomial](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#example-ex-36697a)
- [Solvability by Radicals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#solvability-by-radicals)
- [`SolveByRadicals(f): RngUPolElt → FldFun, [FldFunElt], [FldFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-solvebyradicals-rngupolelt)
- [`CyclicToRadical(K, a, z): FldFun, FldFunElt, RngElt → FldFun, [FldFunElt], [FldFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#function-cyclictoradical-fldfun-fldfunelt-rngelt)
- [Example: Solving a Polynomial by Radicals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/galois.md#example-ex-0a9a35)
- [Subfields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/subfields.md)
- [`Subfields(F): FldFun → SeqEnum[FldFun]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/subfields.md#function-subfields-fldfun)
- [Example: Subfields of a Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/subfields.md#example-ex-aeb7db)
- [Automorphism Group](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md)
- [Automorphisms over the Base Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#automorphisms-over-the-base-field)
- [`Automorphisms(K, k): FldFun, FldFunG → [Map]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-automorphisms-fldfun-fldfung)
- [`AutomorphismGroup(K, k): FldFun, FldFunG → GrpFP, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-automorphismgroup-fldfun-fldfung)
- [Example: Automorphisms of a Degree 7 Extension](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#example-ex-796d6f)
- [`IsSubfield(K, L): FldFun, FldFun → BoolElt, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-issubfield-fldfun-fldfun)
- [`IsIsomorphicOverQt(K, L): FldFun, FldFun → BoolElt, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-isisomorphicoverqt-fldfun-fldfun)
- [Example: Subfields and Isomorphism Testing](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#example-ex-40fb2d)
- [General Automorphisms](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#general-automorphisms)
- [`Isomorphisms(K, E): FldFunG, FldFunG → [Map]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-isomorphisms-fldfung-fldfung)
- [`IsIsomorphic(K, E): FldFunG, FldFunG → BoolElt, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-isisomorphic-fldfung-fldfung)
- [`Automorphisms(K): FldFunG → [Map]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-automorphisms-fldfung)
- [`Isomorphisms(K,E,p1,p2): FldFunG, FldFunG, PlcFunElt, PlcFunElt → [Map]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-isomorphisms-fldfung-fldfung-plcfunelt-plcfunelt)
- [`Automorphisms(K,p1,p2): FldFunG, PlcFunElt, PlcFunElt → [Map]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-automorphisms-fldfung-plcfunelt-plcfunelt)
- [`AutomorphismGroup(K): FldFunG → GrpFP, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-automorphismgroup-fldfung)
- [`AutomorphismGroup(K,f): FldFunG, Map → Grp, Map, SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-automorphismgroup-fldfung-map)
- [Field Morphisms](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#field-morphisms)
- [`IsMorphism(f): Map → Bool`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-ismorphism-map)
- [`FieldMorphism(f): Map → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-fieldmorphism-map)
- [`IdentityFieldMorphism(F): Fld → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-identityfieldmorphism-fld)
- [`IsIdentity(f): Map → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-isidentity-map)
- [`Equality(f, g): Map, Map → Bool`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-equality-map-map)
- [`HasInverse(f): Map → MonStgElt, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-hasinverse-map)
- [`Composition(f, g): Map, Map → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#function-composition-map-map)
- [Example: Isomorphisms of Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/automorphism-group.md#example-ex-9d9eb5)
- [Global Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md)
- [Functions relative to the Exact Constant Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#functions-relative-to-the-exact-constant-field)
- [`NumberOfPlacesOfDegreeOverExactConstantField(F, m): FldFun, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoverexactconstantfield-fldfun-rngintelt)
- [`NumberOfPlacesDegECF(F, m): FldFun, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesdegecf-fldfun-rngintelt)
- [`NumberOfPlacesOfDegreeOneOverExactConstantField(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneoverexactconstantfield-fldfung)
- [`NumberOfPlacesOfDegreeOneECF(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneecf-fldfung)
- [`NumberOfPlacesOfDegreeOneOverExactConstantField(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneoverexactconstantfield-fldfung-rngintelt)
- [`NumberOfPlacesOfDegreeOneECF(F, m): FldFun, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneecf-fldfun-rngintelt)
- [`SerreBound(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-serrebound-fldfung)
- [`SerreBound(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-serrebound-fldfung-rngintelt)
- [`SerreBound(q, g): RngIntElt, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-serrebound-rngintelt-rngintelt)
- [`IharaBound(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-iharabound-fldfung)
- [`IharaBound(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-iharabound-fldfung-rngintelt)
- [`IharaBound(q, g): RngIntElt, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-iharabound-rngintelt-rngintelt)
- [`NumberOfPlacesOfDegreeOneECFBound(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneecfbound-fldfung)
- [`NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneoverexactconstantfieldbound-fldfung)
- [`NumberOfPlacesOfDegreeOneECFBound(F, m): FldFun, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneecfbound-fldfun-rngintelt)
- [`NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneoverexactconstantfieldbound-fldfung-rngintelt)
- [`NumberOfPlacesOfDegreeOneECFBound(q, g): RngIntElt, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneecfbound-rngintelt-rngintelt)
- [`NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(q, g): RngIntElt, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-numberofplacesofdegreeoneoverexactconstantfieldbound-rngintelt-rngintelt)
- [`LPolynomial(F): FldFunG → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-lpolynomial-fldfung)
- [`LPolynomial(F, m): FldFunG, RngIntElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-lpolynomial-fldfung-rngintelt)
- [`ZetaFunction(F): FldFunG → FldFunRatUElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-zetafunction-fldfung)
- [`ZetaFunction(F, m): FldFunG, RngIntElt → FldFunRatUElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-zetafunction-fldfung-rngintelt)
- [Functions Relative to the Constant Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#functions-relative-to-the-constant-field)
- [`Places(F, m): FldFunG, RngIntElt → SeqEnum[PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-places-fldfung-rngintelt)
- [`HasPlace(F, m): FldFunG, RngIntElt → BoolElt, PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-hasplace-fldfung-rngintelt)
- [`HasRandomPlace(F, m): FldFunG, RngIntElt → BoolElt, PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-hasrandomplace-fldfung-rngintelt)
- [`RandomPlace(F, m): FldFunG, RngIntElt → PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-randomplace-fldfung-rngintelt)
- [Example: Invariants of a Global Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#example-ex-8f580e)
- [Example: Invariants of a Global Relative Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#example-ex-9f3184)
- [Functions related to Class Group](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#functions-related-to-class-group)
- [`UnitRank(O): RngFunOrd → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-unitrank-rngfunord)
- [`UnitGroup(O): RngFunOrd → GrpAb, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-unitgroup-rngfunord)
- [`Regulator(O): RngFunOrd → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-regulator-rngfunord)
- [`PrincipalIdealMap(O): RngFunOrd → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-principalidealmap-rngfunord)
- [Example: Unit Group of an Equation Order](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#example-ex-1591a6)
- [`ClassGroup(F : parameters): FldFunG → GrpAb, Map, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classgroup-fldfung)
- [`ClassGroup(O): RngFunOrd → GrpAb, Map, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classgroup-rngfunord)
- [`ClassGroupExactSequence(O): RngFunOrd → Map, Map, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classgroupexactsequence-rngfunord)
- [`ClassGroupAbelianInvariants(F : parameters): FldFunG → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classgroupabelianinvariants-fldfung)
- [`ClassGroupAbelianInvariants(O): RngFunOrd → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classgroupabelianinvariants-rngfunord)
- [`ClassNumber(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classnumber-fldfung)
- [`ClassNumber(O): RngFunOrd → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classnumber-rngfunord)
- [Example: Class Groups of Relative Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#example-ex-dd2e5e)
- [`GlobalUnitGroup(F): FldFunG → GrpAb, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-globalunitgroup-fldfung)
- [`ClassGroupPRank(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-classgroupprank-fldfung)
- [`HasseWittInvariant(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-hassewittinvariant-fldfung)
- [`IndependentUnits(O): RngFunOrd → SeqEnum[RngFunOrdElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-independentunits-rngfunord)
- [`FundamentalUnits(O): RngFunOrd → SeqEnum[RngFunOrdElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#function-fundamentalunits-rngfunord)
- [Example: Basis and Fundamental Units of a Maximal Order](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/global.md#example-ex-065c59)
- [Structure Predicates](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md)
- [`IsField(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isfield-fldfun)
- [`IsEuclideanDomain(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-iseuclideandomain-fldfun)
- [`IsField(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isfield-rngfunord)
- [`IsPID(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-ispid-fldfun)
- [`IsUFD(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isufd-fldfun)
- [`IsDivisionRing(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isdivisionring-fldfun)
- [`IsEuclideanRing(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-iseuclideanring-fldfun)
- [`IsDivisionRing(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isdivisionring-rngfunord)
- [`IsPrincipalIdealRing(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isprincipalidealring-fldfun)
- [`IsDomain(R): FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isdomain-fldfun)
- [`IsDomain(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isdomain-rngfunord)
- [`F eq G: FldFunG, FldFunG → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#operation-op-eq-fldfung-fldfung)
- [`F ne G: FldFunG, FldFunG → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#operation-op-ne-fldfung-fldfung)
- [`O1 eq O2: RngFunOrd, Rng → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#operation-op-eq-rngfunord-rng)
- [`O1 ne O2: RngFunOrd, Rng → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#operation-op-ne-rngfunord-rng)
- [`O1 subset O2: RngFunOrd, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#operation-op-subset-rngfunord-rngfunord)
- [`IsGlobal(F): FldFunG → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isglobal-fldfung)
- [`IsRationalFunctionField(F): FldFunG → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isrationalfunctionfield-fldfung)
- [`IsFiniteOrder(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isfiniteorder-rngfunord)
- [`IsEquationOrder(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isequationorder-rngfunord)
- [`IsAbsoluteOrder(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isabsoluteorder-rngfunord)
- [`IsMaximal(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-ismaximal-rngfunord)
- [`IsTamelyRamified(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-istamelyramified-rngfunord)
- [`IsTotallyRamified(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-istotallyramified-rngfunord)
- [`IsUnramified(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isunramified-rngfunord)
- [`IsWildlyRamified(O): RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-iswildlyramified-rngfunord)
- [`IsInKummerRepresentation(K): FldFun → BoolElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isinkummerrepresentation-fldfun)
- [`IsInArtinSchreierRepresentation(K): FldFun → BoolElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/structure-predicates.md#function-isinartinschreierrepresentation-fldfun)
- [Homomorphisms](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md)
- [`hom R | g>: FldFun, Rng, RngElt → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#constructor-constructor-hom-fldfun-rng-rngelt-map)
- [`hom R | cf, g>: FldFun, Rng, Map, RngElt → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#constructor-constructor-hom-fldfun-rng-map-rngelt-map)
- [`hom< O -> R | g >: RngFunOrd, Rng, RngElt → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#constructor-constructor-hom-rngfunord-rng-rngelt-map)
- [`hom< O -> R | cf, g >: RngFunOrd, Rng, Map, RngElt → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#constructor-constructor-hom-rngfunord-rng-map-rngelt-map)
- [`IsRingHomomorphism(m): Map → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#function-isringhomomorphism-map)
- [Example: Homomorphisms of Relative Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#example-ex-1107ee)
- [`hom< O -> R | b₁, ..., bₙ >: RngFunOrd, Rng, RngElt, ..., RngElt → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#constructor-constructor-hom-rngfunord-rng-rngelt-rngelt-map)
- [`hom< O -> R | m, b₁, ..., bₙ >: RngFunOrd, Rng, Map, RngElt, ..., RngElt → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/hom.md#constructor-constructor-hom-rngfunord-rng-map-rngelt-rngelt-map)
- [Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md)
- [Creation of Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#creation-of-elements)
- [`F . 1: FldFun → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-fldfun-fldfunelt)
- [`F . 2: FldFun → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-fldfun-fldfunelt-2)
- [`Name(F, i): FldFun, RngIntElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-name-fldfun-rngintelt)
- [`O . i: RngFunOrd, RngIntElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-rngfunord-rngintelt-fldfunordelt)
- [`FF . i: FldFunOrd, RngIntElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-fldfunord-rngintelt-fldfunordelt)
- [`F ! a: FldFun, . → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-fldfun)
- [`elt: FldFun, . → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#constructor-constructor-elt-fldfun-fldfunelt)
- [`O ! a: RngFunOrd, . → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-rngfunord)
- [`elt: RngFunOrd, . → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#constructor-constructor-elt-rngfunord-rngfunordelt)
- [`FF ! a: FldFunOrd, Any → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-fldfunord-any)
- [`elt: FldFunOrd, Any → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#constructor-constructor-elt-fldfunord-any-fldfunordelt)
- [`elt< F | a₀, a₁, ..., aₙ₋₁>: FldFun, RngElt, ..., RngElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#constructor-constructor-elt-fldfun-rngelt-rngelt-fldfunelt)
- [`elt< O | a₁, a₂, ..., aₙ>: RngFunOrd, RngElt, ..., RngElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#constructor-constructor-elt-rngfunord-rngelt-rngelt-rngfunordelt)
- [`elt< FF | a₁, a₂, ..., aₙ>: FldFunOrd, RngElt, ..., RngElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#constructor-constructor-elt-fldfunord-rngelt-rngelt-fldfunordelt)
- [`One(F): FldFun → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-one-fldfun)
- [`One(O): RngFunOrd → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-one-rngfunord)
- [`Identity(F): FldFun → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-identity-fldfun)
- [`Identity(O): RngFunOrd → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-identity-rngfunord)
- [`Zero(F): FldFun → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zero-fldfun)
- [`Zero(O): RngFunOrd → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zero-rngfunord)
- [`Representative(F): FldFun → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-representative-fldfun)
- [`Representative(O): RngFunOrd → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-representative-rngfunord)
- [`Random(F, m): FldFunG, RngIntElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-random-fldfung-rngintelt)
- [`Random(O, m): RngFunOrd, RngIntElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-random-rngfunord-rngintelt)
- [Parent and Category](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#parent-and-category)
- [`Parent(a): FldFunElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-parent-fldfunelt)
- [`Parent(a): RngFunOrdElt → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-parent-rngfunordelt)
- [`Category(a): FldFunElt → Cat`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-category-fldfunelt)
- [`Category(a): RngFunOrdElt → Cat`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-category-rngfunordelt)
- [Sequence Conversions](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#sequence-conversions)
- [`ElementToSequence(a): FldFunElt → SeqEnum[FldElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-elementtosequence-fldfunelt)
- [`ElementToSequence(a): RngFunOrdElt → SeqEnum[RngElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-elementtosequence-rngfunordelt)
- [`ElementToSequence(a): FldFunOrdElt → SeqEnum[RngElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-elementtosequence-fldfunordelt)
- [`Eltseq(a): FldFunElt → SeqEnum[FldElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-eltseq-fldfunelt)
- [`Eltseq(a): RngFunOrdElt → SeqEnum[RngElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-eltseq-rngfunordelt)
- [`Eltseq(a): FldFunOrdElt → SeqEnum[RngElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-eltseq-fldfunordelt)
- [`Eltseq(a, R): FldFunElt, FldFunG → [FldFunGElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-eltseq-fldfunelt-fldfung)
- [`Flat(a): FldFunElt → [FldFunGElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-flat-fldfunelt)
- [`F ! [ a₀, a₁, ..., aₙ₋₁ ]: FldFun, SeqEnum → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-fldfun-seqenum)
- [`O ! [ a₁, a₂, ..., aₙ ]: RngFunOrd, SeqEnum → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-rngfunord-seqenum)
- [Example: Creating Function Field Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#example-ex-27bc08)
- [Arithmetic Operators](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#arithmetic-operators)
- [`+ a: FldFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-fldfunelt-fldfunelt)
- [`+ a: RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-rngfunordelt-rngfunordelt)
- [`+ a: FldFunOrdElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-fldfunordelt-fldfunordelt)
- [`- a: FldFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-fldfunelt-fldfunelt-2)
- [`- a: RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-rngfunordelt-rngfunordelt-2)
- [`- a: FldFunOrdElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-fldfunordelt-fldfunordelt-2)
- [`a + b: FldFunElt, FldFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-plus-fldfunelt-fldfunelt)
- [`a + b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-plus-rngfunordelt-rngfunordelt)
- [`a + b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-plus-fldfunordelt-fldfunordelt)
- [`a - b: FldFunElt, FldFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-minus-fldfunelt-fldfunelt)
- [`a - b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-minus-rngfunordelt-rngfunordelt)
- [`a - b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-minus-fldfunordelt-fldfunordelt)
- [`a * b: FldFunElt, FldFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-times-fldfunelt-fldfunelt)
- [`a * b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-times-rngfunordelt-rngfunordelt)
- [`a * b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-times-fldfunordelt-fldfunordelt)
- [`a div b: FldFunElt, FldFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-div-fldfunelt-fldfunelt-fldfunelt)
- [`a div b: FldFunElt, RngElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-div-fldfunelt-rngelt-fldfunelt)
- [`a div b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-div-rngfunordelt-rngfunordelt-rngfunordelt)
- [`a div b: RngFunOrdElt, RngElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-div-rngfunordelt-rngelt-rngfunordelt)
- [`a / b: FldFunElt, FldFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-div-fldfunelt-fldfunelt)
- [`a / b: RngFunOrdElt, RngFunOrdElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-div-rngfunordelt-rngfunordelt)
- [`a / b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-div-fldfunordelt-fldfunordelt)
- [`a ^ k: FldFunElt, RngIntElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-pow-fldfunelt-rngintelt)
- [`a ^ k: RngFunOrdElt, RngIntElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-pow-rngfunordelt-rngintelt)
- [`a ^ k: FldFunOrdElt, RngIntElt → FldFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-pow-fldfunordelt-rngintelt)
- [`Modexp(a, k, m): RngFunOrdElt, RngIntElt, RngUPolElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-modexp-rngfunordelt-rngintelt-rngupolelt)
- [`Modexp(a, k, m): RngFunOrdElt, RngIntElt, RngValElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-modexp-rngfunordelt-rngintelt-rngvalelt)
- [`a mod I: RngFunOrdElt, RngFunOrdIdl → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-operation-mod-rngfunordelt-rngfunordidl-rngfunordelt)
- [`Modinv(a, m): RngFunOrdElt, RngFunOrdIdl → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-modinv-rngfunordelt-rngfunordidl)
- [`Modinv(a, m): RngFunOrdElt, RngUPolElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-modinv-rngfunordelt-rngupolelt)
- [`Modinv(a, m): RngFunOrdElt, RngValElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-modinv-rngfunordelt-rngvalelt)
- [Equality and Membership](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#equality-and-membership)
- [`a eq b: FldFunElt, FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-eq-fldfunelt-fldfunelt)
- [`a eq b: RngFunOrdElt, RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-eq-rngfunordelt-rngfunordelt)
- [`a eq b: FldFunOrdElt, FldFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-eq-fldfunordelt-fldfunordelt)
- [`a ne b: FldFunElt, FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-ne-fldfunelt-fldfunelt)
- [`a ne b: RngFunOrdElt, RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-ne-rngfunordelt-rngfunordelt)
- [`a ne b: FldFunOrdElt, FldFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-ne-fldfunordelt-fldfunordelt)
- [`a in F: RngElt, FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-in-rngelt-fldfun)
- [`a in O: RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-in-rngelt-rngfunord)
- [`a in FF: RngElt, FldFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-in-rngelt-fldfunord)
- [`a notin F: RngElt, FldFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-notin-rngelt-fldfun)
- [`a notin O: RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-notin-rngelt-rngfunord)
- [`a notin FF: RngElt, FldFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#operation-op-notin-rngelt-fldfunord)
- [Predicates on Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#predicates-on-elements)
- [`IsDivisibleBy(a, b): FldFunElt, FldFunElt → BoolElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isdivisibleby-fldfunelt-fldfunelt)
- [`IsDivisibleBy(a, b): RngFunOrdElt, RngFunOrdElt → BoolElt, RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isdivisibleby-rngfunordelt-rngfunordelt)
- [`IsZero(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-iszero-fldfunelt)
- [`IsZero(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-iszero-rngfunordelt)
- [`IsZero(a): FldFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-iszero-fldfunordelt)
- [`IsOne(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isone-fldfunelt)
- [`IsOne(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isone-rngfunordelt)
- [`IsOne(a): FldFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isone-fldfunordelt)
- [`IsMinusOne(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isminusone-fldfunelt)
- [`IsMinusOne(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isminusone-rngfunordelt)
- [`IsMinusOne(a): FldFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isminusone-fldfunordelt)
- [`IsNilpotent(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isnilpotent-fldfunelt)
- [`IsNilpotent(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isnilpotent-rngfunordelt)
- [`IsIdempotent(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isidempotent-fldfunelt)
- [`IsIdempotent(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isidempotent-rngfunordelt)
- [`IsUnit(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isunit-fldfunelt)
- [`IsUnit(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isunit-rngfunordelt)
- [`IsUnit(a): FldFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isunit-fldfunordelt)
- [`IsZeroDivisor(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-iszerodivisor-fldfunelt)
- [`IsZeroDivisor(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-iszerodivisor-rngfunordelt)
- [`IsRegular(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isregular-fldfunelt)
- [`IsRegular(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isregular-rngfunordelt)
- [`IsIrreducible(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isirreducible-fldfunelt)
- [`IsIrreducible(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isirreducible-rngfunordelt)
- [`IsPrime(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isprime-fldfunelt)
- [`IsPrime(a): RngFunOrdElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isprime-rngfunordelt)
- [`IsSeparating(a): FldFunGElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isseparating-fldfungelt)
- [`IsConstant(a): FldFunGElt → BoolElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isconstant-fldfungelt)
- [`IsConstant(a): RngFunOrdElt → BoolElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-isconstant-rngfunordelt)
- [`IsGlobalUnit(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-field-elements-isglobalunit)
- [`IsGlobalUnitWithPreimage(a): FldFunElt → BoolElt, GrpAbElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-field-elements-isglobalunitwithpreimage)
- [`IsUnitWithPreimage(a): RngFunOrdElt → BoolElt, GrpAbElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-orders-elements-isunitwithpreimage)
- [Functions related to Norm and Trace](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#functions-related-to-norm-and-trace)
- [`Trace(a): FldFunGElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-trace-fldfungelt)
- [`Trace(a): RngFunOrdElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-trace-rngfunordelt)
- [`Norm(a): FldFunGElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-norm-fldfungelt)
- [`Norm(a): RngFunOrdElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-norm-rngfunordelt)
- [`MinimalPolynomial(a): FldFunElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-minimalpolynomial-fldfunelt)
- [`MinimalPolynomial(a): RngFunOrdElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-minimalpolynomial-rngfunordelt)
- [`MinimalPolynomial(a): FldFunOrdElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-minimalpolynomial-fldfunordelt)
- [`CharacteristicPolynomial(a): FldFunElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-characteristicpolynomial-fldfunelt)
- [`CharacteristicPolynomial(a): RngFunOrdElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-characteristicpolynomial-rngfunordelt)
- [`RepresentationMatrix(a): FldFunGElt → AlgMatElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-representationmatrix-fldfungelt)
- [`RepresentationMatrix(a): RngFunOrdElt → AlgMatElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-representationmatrix-rngfunordelt)
- [`Trace(a, R): FldFunElt, Rng → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-trace-fldfunelt-rng)
- [`Trace(a, R): RngFunOrdElt, Rng → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-trace-rngfunordelt-rng)
- [`Norm(a, R): FldFunElt, Rng → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-norm-fldfunelt-rng)
- [`Norm(a, R): RngFunOrdElt, Rng → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-norm-rngfunordelt-rng)
- [`CharacteristicPolynomial(a, R): FldFunElt, Rng → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-characteristicpolynomial-fldfunelt-rng)
- [`CharacteristicPolynomial(a, R): RngFunOrdElt, Rng → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-characteristicpolynomial-rngfunordelt-rng)
- [`MinimalPolynomial(a, R): FldFunElt, Rng → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-minimalpolynomial-fldfunelt-rng)
- [`MinimalPolynomial(a, R): RngFunOrdElt, Rng → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-minimalpolynomial-rngfunordelt-rng)
- [`AbsoluteMinimalPolynomial(a): FldFunElt → RngUPolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-absoluteminimalpolynomial-fldfunelt)
- [`RepresentationMatrix(a, R): FldFunGElt, Rng → AlgMatElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-representationmatrix-fldfungelt-rng)
- [`RepresentationMatrix(a, R): RngFunOrdElt, Rng → AlgMatElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-representationmatrix-rngfunordelt-rng)
- [Example: Norm and Trace of Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#example-ex-2d04b2)
- [Functions related to Orders and Integrality](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#functions-related-to-orders-and-integrality)
- [`IntegralSplit(a, O): FldFunElt, RngFunOrd → RngFunOrdElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-integralsplit-fldfunelt-rngfunord)
- [`IntegralSplit(a, O): RngFunOrdElt, RngFunOrd → RngFunOrdElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-integralsplit-rngfunordelt-rngfunord)
- [`IntegralSplit(a, O): FldFunOrdElt, RngFunOrd → RngFunOrdElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-integralsplit-fldfunordelt-rngfunord)
- [`Numerator(a, O): FldFunElt, RngFunOrd → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-numerator-fldfunelt-rngfunord)
- [`Numerator(a): FldFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-numerator-fldfunordelt)
- [`Numerator(a, O): FldFunOrdElt, RngFunOrd → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-numerator-fldfunordelt-rngfunord)
- [`Denominator(a, O): FldFunElt, RngFunOrd → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-denominator-fldfunelt-rngfunord)
- [`Denominator(a): FldFunOrdElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-denominator-fldfunordelt)
- [`Denominator(a, O): FldFunOrdElt, RngFunOrd → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-denominator-fldfunordelt-rngfunord)
- [`Min(a, O): FldFunElt, RngFunOrd → RngElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-min-fldfunelt-rngfunord)
- [`Minimum(a, O): FldFunElt, RngFunOrd → RngElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-minimum-fldfunelt-rngfunord)
- [`Minimum(a, O): RngFunOrdElt, RngFunOrd → RngElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-minimum-rngfunordelt-rngfunord)
- [`Min(a, O): RngFunOrdElt, RngFunOrd → RngElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-min-rngfunordelt-rngfunord)
- [Functions related to Places and Divisors](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#functions-related-to-places-and-divisors)
- [`Evaluate(a, P): FldFunElt, PlcFunElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-evaluate-fldfunelt-plcfunelt)
- [`Evaluate(a, P): RngFunOrdElt, PlcFunElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-evaluate-rngfunordelt-plcfunelt)
- [`Lift(a, P): RngElt, PlcFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-lift-rngelt-plcfunelt)
- [`Valuation(a, P): FldFunElt, PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-valuation-fldfunelt-plcfunelt)
- [`Valuation(a, P): RngFunOrdElt, PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-valuation-rngfunordelt-plcfunelt)
- [`Expand(a, P): FldFunGElt, PlcFunElt → RngSerElt, FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-expand-fldfungelt-plcfunelt)
- [`Development(a, P): FldFunGElt, PlcFunElt → RngSerElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-development-fldfungelt-plcfunelt)
- [`Divisor(a): FldFunGElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-divisor-fldfungelt)
- [`Divisor(a): RngFunOrdElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-divisor-rngfunordelt)
- [`PrincipalDivisor(a): FldFunGElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-principaldivisor-fldfungelt)
- [`PrincipalDivisor(a): RngFunOrdElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-principaldivisor-rngfunordelt)
- [`Zeros(a): FldFunGElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zeros-fldfungelt)
- [`Zeros(a): RngFunOrdElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zeros-rngfunordelt)
- [`Zeroes(a): FldFunGElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zeroes-fldfungelt)
- [`Zeroes(a): RngFunOrdElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zeroes-rngfunordelt)
- [`Zeros(F, a): FldFunG, FldFunGElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zeros-fldfung-fldfungelt)
- [`Zeroes(F, a): FldFunG, FldFunGElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-zeroes-fldfung-fldfungelt)
- [`Poles(a): FldFunGElt → SeqEnum[PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-poles-fldfungelt)
- [`Poles(a): RngFunOrdElt → SeqEnum[PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-poles-rngfunordelt)
- [`Poles(F, a): FldFun, FldFunGElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-poles-fldfun-fldfungelt)
- [`Degree(a): FldFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-degree-fldfunelt)
- [`Degree(a): RngFunOrdElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-degree-rngfunordelt)
- [`CommonZeros(L): [FldFunGElt] → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-commonzeros-fldfungelt)
- [`CommonZeros(F, L): FldFunG, SeqEnum[ FldFunGElt ] → SeqEnum[ PlcFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-commonzeros-fldfung-seqenum-fldfungelt)
- [Example: Minimal Polynomial and Representation Matrix](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#example-ex-0c032c)
- [`Module(L, R): SeqEnum[ FldFunGElt ], Rng → Mod, Map, SeqEnum[ ModElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-module-seqenum-fldfungelt-rng)
- [`Relations(L, R): SeqEnum[ FldFunElt ], Rng → ModTupRng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-relations-seqenum-fldfunelt-rng)
- [`Relations(L, R, m): SeqEnum[ FldFunElt ], Rng, RngIntElt → ModTupRng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-relations-seqenum-fldfunelt-rng-rngintelt)
- [`Roots(f, D): RngUPolElt, DivFunElt → SeqEnum[ FldFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-roots-rngupolelt-divfunelt)
- [Example: Module and Relations over Relative Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#example-ex-7f09fd)
- [Other Operations on Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#other-operations-on-elements)
- [`ProductRepresentation(a): FldFunGElt → [FldFunGElt], [RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-productrepresentation-fldfungelt)
- [`ProductRepresentation(a): RngFunOrdElt → [RngElt], [RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-productrepresentation-rngfunordelt)
- [`ProductRepresentation(Q, S): [FldFunGElt], [RngIntElt] → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-productrepresentation-fldfungelt-rngintelt)
- [`PowerProduct(Q, S): [FldFunGElt], [RngIntElt] → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-powerproduct-fldfungelt-rngintelt)
- [`RationalFunction(a): FldFunGElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-rationalfunction-fldfungelt)
- [`RationalFunction(a, R): FldFunGElt, Rng → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-rationalfunction-fldfungelt-rng)
- [`Differentiation(x, a): FldFunGElt, FldFunGElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-differentiation-fldfungelt-fldfungelt)
- [`Differentiation(x, n, a): FldFunGElt, RngIntElt, FldFunGElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-fldfungelt-differentiation)
- [`DifferentiationSequence(x, n, a): FldFunGElt, RngIntElt, FldFunGElt → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-differentiationsequence-fldfungelt-rngintelt-fldfungelt)
- [`PrimePowerRepresentation(x, k, a): FldFunGElt, RngIntElt, FldFunGElt → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-primepowerrepresentation-fldfungelt-rngintelt-fldfungelt)
- [`Different(a): RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-different-rngfunordelt)
- [`RationalReconstruction(e, f): FldFunElt, RngUPolElt → BoolElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-rationalreconstruction-fldfunelt-rngupolelt)
- [`CoefficientHeight(a): RngFunOrdElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-coefficientheight-rngfunordelt)
- [`CoefficientHeight(a): FldFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-coefficientheight-fldfunelt)
- [`CoefficientLength(a): RngFunOrdElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-coefficientlength-rngfunordelt)
- [`CoefficientLength(a): FldFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#function-coefficientlength-fldfunelt)
- [Example: Product Representation of Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/elements.md#example-ex-d8f7df)
- [Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md)
- [Creation of Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#creation-of-ideals)
- [`ideal< O | a₁, a₂, ... , aₘ >: RngFunOrd, RngElt, ..., RngElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-ideal-rngfunord-rngelt-rngelt-rngfunordidl)
- [`ideal< O | T, d >: RngFunOrd, AlgMatElt, RngElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-ideal-rngfunord-algmatelt-rngelt-rngfunordidl)
- [`ideal< O | T, d >: RngFunOrd, ModDed, RngElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-ideal-rngfunord-modded-rngelt-rngfunordidl)
- [`ideal< O | T, S >: RngFunOrd, AlgMatElt, [RngFunOrdIdl] → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-ideal-rngfunord-algmatelt-rngfunordidl-rngfunordidl)
- [`ideal< O | T, I₁, ..., Iₙ >: RngFunOrd, AlgMatElt, RngFunOrdIdl, ..., RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-ideal-rngfunord-algmatelt-rngfunordidl-rngfunordidl-rngfunordidl)
- [`x * O: RngElt, RngFunOrd → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-times-rngelt-rngfunord)
- [`O * x: RngFunOrd, RngElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-times-rngfunord-rngelt)
- [`Ideal(P): PlcFunElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-ideal-plcfunelt)
- [`Ideals(D): DivFunElt → RngFunOrdIdl, RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-ideals-divfunelt)
- [`O !! I: RngFunOrd, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-rngfunord-rngfunordidl)
- [Parent and Category](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#parent-and-category)
- [`Parent(I): RngFunOrdIdl → PowIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-parent-rngfunordidl)
- [`Category(I): RngFunOrdIdl → Cat`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-category-rngfunordidl)
- [Arithmetic Operators](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#arithmetic-operators)
- [`I + J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-plus-rngfunordidl-rngfunordidl)
- [`I * J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-times-rngfunordidl-rngfunordidl)
- [`I / J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-div-rngfunordidl-rngfunordidl)
- [`I ^ k: RngFunOrdIdl, RngIntElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-pow-rngfunordidl-rngintelt)
- [`c * I: RngElt, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-times-rngelt-rngfunordidl)
- [`I * c: RngFunOrdIdl, RngElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-times-rngfunordidl-rngelt)
- [`I / c: RngFunOrdIdl, RngElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-div-rngfunordidl-rngelt)
- [`c / I: RngElt, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-div-rngelt-rngfunordidl)
- [`IdealQuotient(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-idealquotient-rngfunordidl-rngfunordidl)
- [`ColonIdeal(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-colonideal-rngfunordidl-rngfunordidl)
- [`ChineseRemainderTheorem(I1, I2, e1, e2): RngFunOrdIdl, RngFunOrdIdl, RngFunOrdElt, RngFunOrdElt → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-chineseremaindertheorem-rngfunordidl-rngfunordidl-rngfunordelt-rngfunordelt)
- [`CRT(I1, I2, e1, e2): RngFunOrdIdl, RngFunOrdIdl, RngFunOrdElt, RngFunOrdElt → RngOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-crt-rngfunordidl-rngfunordidl-rngfunordelt-rngfunordelt)
- [Roots of Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#roots-of-ideals)
- [`IsPower(I, n): RngFunOrdIdl, RngIntElt → BoolElt, RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-ispower-rngfunordidl-rngintelt)
- [`Root(I, n): RngFunOrdIdl, RngIntElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-root-rngfunordidl-rngintelt)
- [`IsSquare(I): RngFunOrdIdl → BoolElt, RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-issquare-rngfunordidl)
- [`SquareRoot(I): RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-squareroot-rngfunordidl)
- [`Sqrt(I): RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-sqrt-rngfunordidl)
- [Example: Testing Whether an Ideal Is a Square](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#example-ex-eaf22a)
- [Equality and Membership](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#equality-and-membership)
- [`I eq J: RngFunOrdIdl, RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-eq-rngfunordidl-rngfunordidl)
- [`I ne J: RngFunOrdIdl, RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-ne-rngfunordidl-rngfunordidl)
- [`I in S: RngFunOrdIdl, PowIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-in-rngfunordidl-powidl)
- [`I notin S: RngFunOrdIdl, PowIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-notin-rngfunordidl-powidl)
- [Predicates on Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#predicates-on-ideals)
- [`IsZero(I): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-iszero-rngfunordidl)
- [`IsOne(I): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isone-rngfunordidl)
- [`IsIntegral(I): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isintegral-rngfunordidl)
- [`IsPrime(I): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isprime-rngfunordidl)
- [`IsPrincipal(I): RngFunOrdIdl → BoolElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isprincipal-rngfunordidl)
- [Predicates on Prime Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#predicates-on-prime-ideals)
- [`IsInert(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isinert-rngfunordidl)
- [`IsInert(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isinert-rngfunordidl-rngfunord)
- [`IsInert(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isinert-rngelt-rngfunord)
- [`IsRamified(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isramified-rngfunordidl)
- [`IsRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isramified-rngfunordidl-rngfunord)
- [`IsRamified(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isramified-rngelt-rngfunord)
- [`IsSplit(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-issplit-rngfunordidl)
- [`IsSplit(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-issplit-rngfunordidl-rngfunord)
- [`IsSplit(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-issplit-rngelt-rngfunord)
- [`IsTamelyRamified(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istamelyramified-rngfunordidl)
- [`IsTamelyRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istamelyramified-rngfunordidl-rngfunord)
- [`IsTamelyRamified(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istamelyramified-rngelt-rngfunord)
- [`IsTotallyRamified(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istotallyramified-rngfunordidl)
- [`IsTotallyRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istotallyramified-rngfunordidl-rngfunord)
- [`IsTotallyRamified(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istotallyramified-rngelt-rngfunord)
- [`IsTotallySplit(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istotallysplit-rngfunordidl)
- [`IsTotallySplit(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istotallysplit-rngfunordidl-rngfunord)
- [`IsTotallySplit(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-istotallysplit-rngelt-rngfunord)
- [`IsUnramified(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isunramified-rngfunordidl)
- [`IsUnramified(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isunramified-rngfunordidl-rngfunord)
- [`IsUnramified(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isunramified-rngelt-rngfunord)
- [`IsWildlyRamified(P): RngFunOrdIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-iswildlyramified-rngfunordidl)
- [`IsWildlyRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-iswildlyramified-rngfunordidl-rngfunord)
- [`IsWildlyRamified(P, O): RngElt, RngFunOrd → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-iswildlyramified-rngelt-rngfunord)
- [Further Ideal Operations](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#further-ideal-operations)
- [`I meet J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-meet-rngfunordidl-rngfunordidl)
- [`Gcd(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-gcd-rngfunordidl-rngfunordidl)
- [`Lcm(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-lcm-rngfunordidl-rngfunordidl)
- [`Factorization(I): RngFunOrdIdl → [ ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-factorization-rngfunordidl)
- [`Factorisation(I): RngFunOrdIdl → []`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-factorisation-rngfunordidl)
- [`Decomposition(O, p): RngFunOrd, RngElt → [ RngFunOrdIdl ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-decomposition-rngfunord-rngelt)
- [`Decomposition(O, p): RngFunOrd, RngFunOrdIdl → [ RngFunOrdIdl ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-decomposition-rngfunord-rngfunordidl)
- [`Decomposition(O): RngFunOrd → [ RngFunOrdIdl ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-decomposition-rngfunord)
- [`DecompositionType(O, p): RngFunOrd, RngElt → [ ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-decompositiontype-rngfunord-rngelt)
- [`DecompositionType(O, p): RngFunOrd, RngFunOrdIdl → [ ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-decompositiontype-rngfunord-rngfunordidl)
- [`DecompositionType(O): RngFunOrd → [ ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-decompositiontype-rngfunord)
- [`MultiplicatorRing(I): RngFunOrdIdl → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-multiplicatorring-rngfunordidl-2)
- [`pMaximalOrder(O, p): RngFunOrd, RngFunOrdIdl → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-pmaximalorder-rngfunord-rngfunordidl-2)
- [`pRadical(O, p): RngFunOrd, RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-pradical-rngfunord-rngfunordidl-2)
- [`Valuation(a, P): RngElt, RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-valuation-rngelt-rngfunordidl)
- [`Valuation(I, P): RngFunOrdIdl, RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-valuation-rngfunordidl-rngfunordidl)
- [`Order(I): RngFunOrdIdl → RngFunOrd`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-order-rngfunordidl)
- [`Denominator(I): RngFunOrdIdl → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-denominator-rngfunordidl)
- [`Minimum(I): RngFunOrdIdl → Any`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-minimum-rngfunordidl)
- [`I meet R: RngFunOrdIdl, Rng → Any`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-meet-rngfunordidl-rng)
- [`IntegralSplit(I): RngFunOrdIdl → RngFunOrdIdl, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-integralsplit-rngfunordidl)
- [`Norm(I): RngFunOrdIdl → Any`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-norm-rngfunordidl)
- [`TwoElement(I): RngFunOrdIdl → RngElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-twoelement-rngfunordidl)
- [`Generators(I): RngFunOrdIdl → [ RngFunOrdElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-generators-rngfunordidl)
- [`Basis(I): RngFunOrdIdl → [FldFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-basis-rngfunordidl)
- [`Basis(I, R): RngFunOrdIdl, Rng → [ RngElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-basis-rngfunordidl-rng)
- [`BasisMatrix(I): RngFunOrdIdl → AlgMatElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-basismatrix-rngfunordidl)
- [`TransformationMatrix(I): RngFunOrdIdl → AlgMatElt, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-transformationmatrix-rngfunordidl)
- [`CoefficientIdeals(I): RngFunOrdIdl → [RngFunOrdIdl]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-coefficientideals-rngfunordidl)
- [`Different(I): RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-different-rngfunordidl)
- [`Codifferent(I): RngFunOrdIdl → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-codifferent-rngfunordidl)
- [`Divisor(I): RngFunOrdIdl → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-divisor-rngfunordidl)
- [`Divisor(I, J): RngFunOrdIdl, RngFunOrdIdl → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-divisor-rngfunordidl-rngfunordidl)
- [`CoefficientHeight(I): RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-coefficientheight-rngfunordidl)
- [`CoefficientLength(I): RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-coefficientlength-rngfunordidl)
- [Example: Ideal Operations in a Relative Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#example-ex-bdacf7)
- [Functions on Prime Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#functions-on-prime-ideals)
- [`RamificationIndex(I): RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-ramificationindex-rngfunordidl)
- [`RamificationDegree(I): RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-ramificationdegree-rngfunordidl)
- [`Degree(I): RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-degree-rngfunordidl)
- [`InertiaDegree(I): RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-inertiadegree-rngfunordidl)
- [`ResidueClassDegree(I): RngFunOrdIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-residueclassdegree-rngfunordidl)
- [`ResidueClassField(I): RngFunOrdIdl → Rng, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-residueclassfield-rngfunordidl)
- [`Place(I): RngFunOrdIdl → PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-place-rngfunordidl)
- [`SafeUniformizer(P): RngFunOrdIdl → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-safeuniformizer-rngfunordidl)
- [`WeakApproximation(I, V): [RngFunOrdIdl], [RngIntElt] → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-weakapproximation-rngfunordidl-rngintelt)
- [Example: Factoring an Ideal of a Maximal Order](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#example-ex-a0157f)
- [Quotient Rings](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#quotient-rings)
- [Operations on Quotient Rings](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operations-on-quotient-rings)
- [`quo< O | I >: RngFunOrd, RngFunOrdIdl → RngFunOrdRes`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-quo-rngfunord-rngfunordidl-rngfunordres)
- [`quo< O | M >: RngFunOrd, ModDed → RngFunOrdRes`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-quo-rngfunord-modded-rngfunordres)
- [`quo< O | M >: RngFunOrd, AlgMatElt → RngFunOrdRes`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-quo-rngfunord-algmatelt-rngfunordres)
- [`quo< O | S >: RngFunOrd, RngElt, ..., RngElt → RngFunOrdRes`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-quo-rngfunord-rngelt-rngelt-rngfunordres)
- [`quo< O | p >: RngFunOrd, RngUPolElt → RngFunOrdRes`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-quo-rngfunord-rngupolelt-rngfunordres)
- [`quo< O | p, r >: RngFunOrd, RngUPolElt, RngIntElt → RngFunOrdRes`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#constructor-constructor-quo-rngfunord-rngupolelt-rngintelt-rngfunordres)
- [`Modulus(OQ): RngFunOrdRes → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-modulus-rngfunordres)
- [Example: Quotient Rings of Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#example-ex-d5cfa1)
- [Elements of Quotients](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#elements-of-quotients)
- [`OQ ! a: RngFunOrdRes, Elt → RngFunOrdResElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-rngfunordres-elt)
- [`a mod I: RngFunOrdElt, RngFunOrdIdl → RngFunOrdElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-operation-mod-rngfunordelt-rngfunordidl-rngfunordelt)
- [`a * b: RngFunOrdResElt, RngFunOrdResElt → RngFunOrdResElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-times-rngfunordreselt-rngfunordreselt)
- [`a + b: RngFunOrdResElt, RngFunOrdResElt → RngFunOrdResElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-plus-rngfunordreselt-rngfunordreselt)
- [`a - b: RngFunOrdResElt, RngFunOrdResElt → RngFunOrdResElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-minus-rngfunordreselt-rngfunordreselt)
- [`a / b: RngFunOrdResElt, RngFunOrdResElt → RngFunOrdResElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-div-rngfunordreselt-rngfunordreselt)
- [`- a: RngFunOrdResElt → RngFunOrdResElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-operation-rngfunordreselt-rngfunordreselt)
- [`a ^ n: RngFunOrdResElt, RngIntElt → RngFunOrdResElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-pow-rngfunordreselt-rngintelt)
- [`a eq b: RngFunOrdResElt, RngFunOrdResElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-eq-rngfunordreselt-rngfunordreselt)
- [`a ne b: RngFunOrdResElt, RngFunOrdResElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#operation-op-ne-rngfunordreselt-rngfunordreselt)
- [`IsZero(a): RngFunOrdResElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-iszero-rngfunordreselt)
- [`IsOne(a): RngFunOrdResElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isone-rngfunordreselt)
- [`IsMinusOne(a): RngFunOrdResElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isminusone-rngfunordreselt)
- [`IsUnit(a): RngFunOrdResElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-isunit-rngfunordreselt)
- [`Eltseq(a): RngFunOrdResElt → []`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-eltseq-rngfunordreselt)
- [`ElementToSequence(a): RngFunOrdResElt → []`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideals.md#function-elementtosequence-rngfunordreselt)
- [Places](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md)
- [Creation of Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#creation-of-structures)
- [`Places(F): FldFun → PlcFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-places-fldfun)
- [Creation of Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#creation-of-elements)
- [General Function Field Places](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#general-function-field-places)
- [`Decomposition(F, P): FldFunG, PlcFunElt → [ PlcFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-fldfung-decomp)
- [`DecompositionType(F, P): FldFun, PlcFunElt → [ ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-decompositiontype-fldfun-plcfunelt)
- [`Zeros(a): FldFunElt → [ PlcFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-zeros-fldfunelt)
- [`Zeros(a): RngFunOrdElt → [ PlcFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-zeros-rngfunordelt-2)
- [`Poles(a): FldFunElt → [ PlcFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-poles-fldfunelt)
- [`Poles(a): RngFunOrdElt → [ PlcFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-poles-rngfunordelt-2)
- [`S ! I: PlcFun, RngFunOrdIdl → PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-plcfun-rngfunordidl)
- [`Place(I): RngFunOrdIdl → PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-place-rngfunordidl-2)
- [`Support(D): DivFunElt → [ PlcFunElt ], [ RngIntElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-support-divfunelt)
- [`Support(P): PlcFunElt → [ PlcFunElt ], [ RngIntElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-support-plcfunelt)
- [`AssignNames(~P, s): PlcFunElt, [ MonStgElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#procedure-assignnames-plcfunelt-monstgelt-ref)
- [`InfinitePlaces(F): FldFun → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-infiniteplaces-fldfun)
- [Global Function Field Places](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#global-function-field-places)
- [`HasPlace(F, m): FldFun, RngIntElt → PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-hasplace-fldfun-rngintelt)
- [`HasRandomPlace(F, m): FldFun, RngIntElt → BoolElt, PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-hasrandomplace-fldfun-rngintelt)
- [`RandomPlace(F, m): FldFun, RngIntElt → PlcFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-randomplace-fldfun-rngintelt)
- [`Places(F, m): FldFun, RngIntElt → SeqEnum[PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-places-fldfun-rngintelt)
- [Example: Creating Places](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#example-ex-92c345)
- [Related Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#related-structures)
- [Parent and Category](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#parent-and-category)
- [`FunctionField(S): PlcFun → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-functionfield-plcfun)
- [`DivisorGroup(F): FldFunG → DivFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-divisorgroup-fldfung)
- [Structure Invariants](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#structure-invariants)
- [General Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#general-function-fields)
- [`WeierstrassPlaces(F): FldFunG → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-weierstrassplaces-fldfung-2)
- [Global Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#global-function-fields)
- [`NumberOfPlacesOfDegreeOneOverExactConstantField(F, m): FldFun, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-numberofplacesofdegreeoneoverexactconstantfield-fldfun-rngintelt)
- [`NumberOfPlacesOfDegreeOneECF(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-numberofplacesofdegreeoneecf-fldfung-rngintelt)
- [`NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F, m): FldFun, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-numberofplacesofdegreeoneoverexactconstantfieldbound-fldfun-rngintelt)
- [`NumberOfPlacesOfDegreeOneECFBound(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-numberofplacesofdegreeoneecfbound-fldfung-rngintelt)
- [`NumberOfPlacesOfDegreeOverExactConstantField(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-numberofplacesofdegreeoverexactconstantfield-fldfung-rngintelt)
- [`NumberOfPlacesDegECF(F, m): FldFunG, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-numberofplacesdegecf-fldfung-rngintelt)
- [Structure Predicates](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#structure-predicates)
- [`S1 eq S2: PlcFun, PlcFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-eq-plcfun-plcfun)
- [`S1 ne S2: PlcFun, PlcFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-ne-plcfun-plcfun)
- [Element Operations](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#element-operations)
- [Parent and Category](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#id2)
- [`Parent(P): PlcFunElt → PlcFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-parent-plcfunelt)
- [`Category(P): PlcFunElt → Cat`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-category-plcfunelt)
- [Arithmetic Operators](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#arithmetic-operators)
- [`- P: PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-operation-plcfunelt-divfunelt)
- [`P1 + P2: PlcFunElt, PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-plus-plcfunelt-plcfunelt)
- [`P1 - P2: PlcFunElt, PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-minus-plcfunelt-plcfunelt)
- [`k * P: RngIntElt, PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-times-rngintelt-plcfunelt)
- [`P div k: PlcFunElt, RngIntElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-operation-div-plcfunelt-rngintelt-divfunelt)
- [`P mod k: PlcFunElt, RngIntElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-operation-mod-plcfunelt-rngintelt-divfunelt)
- [`Quotrem(P, k): PlcFunElt, RngIntElt → DivFunElt, DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-quotrem-plcfunelt-rngintelt)
- [Equality and Membership](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#equality-and-membership)
- [`P1 eq P2: PlcFunElt, PlcFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-eq-plcfunelt-plcfunelt)
- [`P1 ne P2: PlcFunElt, PlcFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-ne-plcfunelt-plcfunelt)
- [`P in S: PlcFunElt, PlcFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-in-plcfunelt-plcfun)
- [`P notin S: PlcFunElt, PlcFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#operation-op-notin-plcfunelt-plcfun)
- [Predicates on Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#predicates-on-elements)
- [`IsFinite(P): PlcFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-isfinite-plcfunelt)
- [`IsWeierstrassPlace(P): PlcFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-isweierstrassplace-plcfunelt)
- [`IsWeierstrassPlace(F, P): FldFunG, PlcFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-isweierstrassplace-fldfung-plcfunelt)
- [Other Element Operations](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#other-element-operations)
- [`FunctionField(P): PlcFunElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-functionfield-plcfunelt)
- [`Degree(P): PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-degree-plcfunelt)
- [`RamificationIndex(P): PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-ramificationindex-plcfunelt)
- [`RamificationDegree(P): PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-ramificationdegree-plcfunelt)
- [`InertiaDegree(P): PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-inertiadegree-plcfunelt)
- [`ResidueClassDegree(P): PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-residueclassdegree-plcfunelt)
- [`Minimum(P): PlcFunElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-minimum-plcfunelt)
- [`ResidueClassField(P): PlcFunElt → Rng, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-residueclassfield-plcfunelt)
- [`Evaluate(a, P): RngElt, PlcFunElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-evaluate-rngelt-plcfunelt)
- [`Lift(a, P): RngElt, PlcFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-lift-rngelt-plcfunelt-2)
- [`Lift(i, P): Infty, PlcFunElt → FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-lift-infty-plcfunelt)
- [`TwoGenerators(P): PlcFunElt → FldFunGElt, FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-twogenerators-plcfunelt)
- [`LocalUniformizer(P): PlcFunElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-fldfung-localuniformizer)
- [`UniformizingElement(P): PlcFunElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-uniformizingelement-plcfunelt)
- [`Valuation(a, P): FldFunElt, PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-valuation-fldfunelt-plcfunelt-2)
- [`Ideal(P): PlcFunElt → RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-ideal-plcfunelt-2)
- [`Norm(P): PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-norm-plcfunelt)
- [Example: Counting Places of a Given Degree](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#example-ex-bbb5e4)
- [Completion at Places](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#completion-at-places)
- [`Completion(F, p): FldFun, PlcFunElt → RngSerLaur, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-completion-fldfun-plcfunelt)
- [`Completion(O, p): RngFunOrd, PlcFunElt → RngSerPow, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/Places.md#function-completion-rngfunord-plcfunelt)
- [Divisors](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md)
- [Creation of Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#creation-of-structures)
- [`DivisorGroup(F): FldFun → DivFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisorgroup-fldfun-2)
- [Creation of Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#creation-of-elements)
- [`Divisor(P): PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisor-plcfunelt)
- [`Div ! P: DivFun, PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-divfun-plcfunelt)
- [`1 * P: PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-operation-plcfunelt-divfunelt)
- [`Div ! a: DivFun, RngElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-divfun-rngelt)
- [`Divisor(a): FldFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisor-fldfunelt)
- [`Divisor(a): RngFunOrdElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisor-rngfunordelt-2)
- [`Div ! I: DivFun, RngFunOrdIdl → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-divfun-rngfunordidl)
- [`Divisor(I): RngFunOrdIdl → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisor-rngfunordidl-2)
- [`Divisor(I, J): RngFunOrdIdl, RngFunOrdIdl → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisor-rngfunordidl-rngfunordidl-2)
- [`Identity(G): DivFun → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-identity-divfun)
- [`Id(G): DivFun → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-id-divfun)
- [`CanonicalDivisor(F): FldFunG → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-canonicaldivisor-fldfung)
- [`DifferentDivisor(F): FldFunG → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-differentdivisor-fldfung)
- [`AssignNames(~D, s): DivFunElt, [ MonStgElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#procedure-assignnames-divfunelt-monstgelt-ref)
- [Related Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#related-structures)
- [Parent and Category](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#parent-and-category)
- [`FunctionField(G): DivFun → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-functionfield-divfun)
- [`Places(F): FldFun → PlcFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-places-fldfun-2)
- [Structure Invariants](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#structure-invariants)
- [`NumberOfSmoothDivisors(n, m, P): RngIntElt, RngIntElt, SeqEnum[RngElt] → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-numberofsmoothdivisors-rngintelt-rngintelt-seqenum-rngelt)
- [`DivisorOfDegreeOne(F): FldFunG → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisorofdegreeone-fldfung)
- [Structure Predicates](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#structure-predicates)
- [`Div1 eq Div2: DivFun, DivFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-eq-divfun-divfun)
- [`Div1 ne Div2: DivFun, DivFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-ne-divfun-divfun)
- [Element Operations](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#element-operations)
- [Arithmetic Operators](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#arithmetic-operators)
- [`- D: DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-operation-divfunelt-divfunelt)
- [`D1 + D2: DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-plus-divfunelt-divfunelt)
- [`D1 - D2: DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-minus-divfunelt-divfunelt)
- [`k * D: RngIntElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-times-rngintelt-divfunelt)
- [`D div k: DivFunElt, RngIntElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-operation-div-divfunelt-rngintelt-divfunelt)
- [`D mod k: DivFunElt, RngIntElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-operation-mod-divfunelt-rngintelt-divfunelt)
- [`P + D: PlcFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-plus-plcfunelt-divfunelt)
- [`D + P: DivFunElt, PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-plus-divfunelt-plcfunelt)
- [`D - P: DivFunElt, PlcFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-minus-divfunelt-plcfunelt)
- [`P - D: PlcFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-minus-plcfunelt-divfunelt)
- [`Quotrem(D, k): DivFunElt, RngIntElt → DivFunElt, DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-quotrem-divfunelt-rngintelt)
- [`GCD(D1, D2): DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-gcd-divfunelt-divfunelt)
- [`Gcd(D1, D2): DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-gcd-divfunelt-divfunelt-2)
- [`GreatestCommonDivisor(D1, D2): DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-greatestcommondivisor-divfunelt-divfunelt)
- [`LCM(D1, D2): DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-lcm-divfunelt-divfunelt)
- [`Lcm(D1, D2): DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-lcm-divfunelt-divfunelt-2)
- [`LeastCommonMultiple(D1, D2): DivFunElt, DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-leastcommonmultiple-divfunelt-divfunelt)
- [Equality, Comparison, and Membership](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#equality-comparison-and-membership)
- [`D1 eq D2: DivFunElt, DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-eq-divfunelt-divfunelt)
- [`D1 ne D2: DivFunElt, DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-ne-divfunelt-divfunelt)
- [`D1 le D2: DivFunElt, DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-le-divfunelt-divfunelt)
- [`D1 lt D2: DivFunElt, DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-lt-divfunelt-divfunelt)
- [`D1 ge D2: DivFunElt, DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-ge-divfunelt-divfunelt)
- [`D1 gt D2: DivFunElt, DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-gt-divfunelt-divfunelt)
- [`D in Div: DivFunElt, DivFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-in-divfunelt-divfun)
- [`D notin Div: DivFunElt, DivFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#operation-op-notin-divfunelt-divfun)
- [Predicates on Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#predicates-on-elements)
- [`IsZero(D): DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-iszero-divfunelt)
- [`IsEffective(D): DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-iseffective-divfunelt)
- [`IsPositive(D): DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-ispositive-divfunelt)
- [`IsSpecial(D): DivFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-isspecial-divfunelt)
- [`IsPrincipal(D): DivFunElt → BoolElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-isprincipal-divfunelt)
- [`IsCanonical(D): DivFunElt → BoolElt, DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-iscanonical-divfunelt)
- [Example: Divisors on a Relative Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#example-ex-749efd)
- [Other Element Operations](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#other-element-operations)
- [`FunctionField(D): DivFunElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-functionfield-divfunelt)
- [`Degree(D): DivFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-degree-divfunelt)
- [`Support(D): DivFunElt → [ PlcFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-support)
- [`Numerator(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-numerator-divfunelt)
- [`ZeroDivisor(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-zerodivisor-divfunelt)
- [`Denominator(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-denominator-divfunelt)
- [`PoleDivisor(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-poledivisor-divfunelt)
- [`Ideals(D): DivFunElt → RngFunOrdIdl, RngFunOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-ideals)
- [`Norm(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-norm-divfunelt)
- [`FiniteSplit(D): DivFunElt → DivFunElt, DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-finitesplit-divfunelt)
- [`FiniteDivisor(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-finitedivisor-divfunelt)
- [`InfiniteDivisor(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-infinitedivisor-divfunelt)
- [`Dimension(D): DivFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-dimension)
- [`IndexOfSpeciality(D): DivFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-indexofspeciality-divfunelt)
- [`ShortBasis(D : parameters): DivFunElt → [RngElt], [RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-shortbasis-divfunelt)
- [`Basis(D : parameters): DivFunElt → [ FldFunElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-basis-divfunelt)
- [`RiemannRochSpace(D): DivFunElt → ModFld, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-riemannrochspace-divfunelt)
- [`RiemannRochSpace(P): PlcFunElt → ModFld, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-riemannrochspace-plcfunelt)
- [`Valuation(D, P): DivFunElt, PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-valuation-divfunelt-plcfunelt)
- [`Reduction(D): DivFunElt → DivFunElt, RngIntElt, DivFunElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-reduction)
- [`Reduction(D, A): DivFunElt, DivFunElt → DivFunElt, RngIntElt, DivFunElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-reduction-divfunelt-divfunelt)
- [`GapNumbers(D, P): DivFunElt, PlcFunElt → SeqEnum[RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-gapnumbers)
- [`GapNumbers(D): DivFunElt → SeqEnum[RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-gapnumbers-divfunelt)
- [Example: Riemann-Roch Space of a Divisor](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#example-ex-bddf6b)
- [Example: Algebraic Relation for Sine and Cosine](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#example-ex-3b8680)
- [Example: Sine and Cosine via Exponentials](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#example-ex-f4fe11)
- [`RamificationDivisor(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-ramificationdivisor)
- [`WeierstrassPlaces(D): DivFunElt → [PlcFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-weierstrassplaces)
- [`IsWeierstrassPlace(D, P): DivFunElt, PlcFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-isweierstrassplace-divfunelt-plcfunelt)
- [`WronskianOrders(D): DivFunElt → [RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-wronskianorders)
- [`ComplementaryDivisor(D): DivFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divfunelt-complementarydivisor)
- [`DifferentialBasis(D): DivFunElt → [DiffFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-differentialbasis-divfunelt)
- [`DifferentialSpace(D): DivFunElt → ModFld, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-differentialspace-divfunelt)
- [`Parametrization(F, D): FldFun, DivFunElt → FldFunElt, [FldFunRatUElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-parametrization-fldfun-divfunelt)
- [Functions related to Divisor Class Groups of Global Function Fields](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#functions-related-to-divisor-class-groups-of-global-function-fields)
- [`ClassGroupGenerationBound(q, g): RngIntElt, RngIntElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-classgroupgenerationbound-rngintelt-rngintelt)
- [`ClassGroupGenerationBound(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-classgroupgenerationbound-fldfung)
- [`ClassNumberApproximation(F, e): FldFunG, FldReElt → FldReElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-classnumberapproximation-fldfung-fldreelt)
- [`ClassNumberApproximationBound(q, g, e): RngIntElt, RngIntElt, FldReElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-classnumberapproximationbound-rngintelt-rngintelt-fldreelt)
- [`ClassGroup(F : parameters): FldFun → GrpAb, Map, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisors-classes-classgroup)
- [`ClassGroupAbelianInvariants(F : parameters): FldFun → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisors-classes-classgroupabelianinvariants)
- [`ClassNumber(F): FldFun → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-classnumber-fldfun)
- [Example: Divisor Class Group Computations](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#example-ex-caa2d2)
- [`GlobalUnitGroup(F): FldFun → GrpAb, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-globalunitgroup-fldfun)
- [`IsGlobalUnit(a): FldFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-isglobalunit-fldfunelt)
- [`IsGlobalUnitWithPreimage(a): FldFunElt → BoolElt, GrpAbElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-isglobalunitwithpreimage-fldfunelt)
- [`PrincipalDivisorMap(F): FldFunG → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-principaldivisormap-fldfung)
- [`ClassGroupExactSequence(F): FldFunG → Map, Map, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-classgroupexactsequence-fldfung)
- [`SUnitGroup(S): SetEnum[PlcFunElt] → GrpAb, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-sunitgroup-setenum-plcfunelt)
- [`IsSUnit(a, S): FldFunElt, SetEnum[PlcFunElt] → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-issunit-fldfunelt-setenum-plcfunelt)
- [`IsSUnitWithPreimage(a, S): FldFunElt, SetEnum[PlcFunElt] → BoolElt, GrpAbElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-issunitwithpreimage-fldfunelt-setenum-plcfunelt)
- [`SRegulator(S): SetEnum[PlcFunElt] → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-sregulator-setenum-plcfunelt)
- [`SPrincipalDivisorMap(S): SetEnum[PlcFunElt] → Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-sprincipaldivisormap-setenum-plcfunelt)
- [`IsSPrincipal(D, S): DivFunElt, SetEnum[PlcFunElt] → BoolElt, FldFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-issprincipal-divfunelt-setenum-plcfunelt)
- [`SClassGroup(S): SetEnum[PlcFunElt] → GrpAb, Map, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-sclassgroup-setenum-plcfunelt)
- [`SClassGroupExactSequence(S): SetEnum[PlcFunElt] → Map, Map, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-sclassgroupexactsequence-setenum-plcfunelt)
- [`SClassGroupAbelianInvariants(S): SetEnum[PlcFunElt] → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-sclassgroupabelianinvariants-setenum-plcfunelt)
- [`SClassNumber(S): SetEnum[PlcFunElt] → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-sclassnumber-setenum-plcfunelt)
- [`ClassGroupPRank(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisors-classes-classgroupprank)
- [`HasseWittInvariant(F): FldFunG → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-divisors-classes-hassewittinvariant)
- [`TateLichtenbaumPairing(D1, D2, m): DivFunElt, DivFunElt, RngIntElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#function-tatelichtenbaumpairing-divfunelt-divfunelt-rngintelt)
- [Example: Tate-Lichtenbaum Pairing of Divisors](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors.md#example-ex-7f9ab0)
- [Differentials](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md)
- [Creation of Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#creation-of-structures)
- [`DifferentialSpace(F): FldFunG → DiffFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-differentialspace-fldfung)
- [Creation of Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#creation-of-elements)
- [`Differential(a): FldFunGElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-differential-fldfungelt)
- [`Differential(a): RngFunOrdElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-differential-rngfunordelt)
- [`Identity(D): DiffFun → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-identity-difffun)
- [`Id(D): DiffFun → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-id-difffun)
- [`IsCanonical(D): DivFunElt → BoolElt, DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-iscanonical-divfunelt-2)
- [Related Structures](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#related-structures)
- [`FunctionField(D): DiffFun → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-functionfield-difffun)
- [`FunctionField(d): DiffFunElt → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-functionfield-difffunelt)
- [Subspaces](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#subspaces)
- [`SpaceOfDifferentialsFirstKind(F): FldFunG → ModFld, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-spaceofdifferentialsfirstkind-fldfung)
- [`SpaceOfHolomorphicDifferentials(F): FldFunG → ModFld, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-spaceofholomorphicdifferentials-fldfung)
- [`BasisOfDifferentialsFirstKind(F): FldFunG → SeqEnum[DiffFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-basisofdifferentialsfirstkind-fldfung)
- [`BasisOfHolomorphicDifferentials(F): FldFunG → SeqEnum[DiffFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-basisofholomorphicdifferentials-fldfung)
- [`DifferentialBasis(D): DivFunElt → [DiffFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-differentials-subspaces-differentialbasis)
- [`DifferentialSpace(D): DivFunElt → ModFld, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-differentialspace-divfunelt-2)
- [Example: Differential Space of a Divisor](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#example-ex-b69721)
- [Structure Predicates](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#structure-predicates)
- [`D1 eq D2: DiffFun, DiffFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-eq-difffun-difffun)
- [Operations on Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operations-on-elements)
- [Arithmetic Operators](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#arithmetic-operators)
- [`r * x: RngElt, DiffFunElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-times-rngelt-difffunelt)
- [`x * r: DiffFunElt, RngElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-times-difffunelt-rngelt)
- [`x + y: DiffFunElt, DiffFunElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-plus-difffunelt-difffunelt)
- [`- x: DiffFunElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-operation-difffunelt-difffunelt)
- [`x - y: DiffFunElt, DiffFunElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-minus-difffunelt-difffunelt)
- [`x / y: DiffFunElt, DiffFunElt → FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-div-difffunelt-difffunelt)
- [`x / r: DiffFunElt, RngElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-div-difffunelt-rngelt)
- [Equality and Membership](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#equality-and-membership)
- [`x eq y: DiffFunElt, DiffFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-eq-difffunelt-difffunelt)
- [`x in D: Any, DiffFun → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#operation-op-in-any-difffun)
- [Predicates on Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#predicates-on-elements)
- [`IsExact(d): DiffFunElt → BoolElt, FldFunGElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-isexact-difffunelt)
- [`IsZero(d): DiffFunElt → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-iszero-difffunelt)
- [Functions on Elements](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#functions-on-elements)
- [`Valuation(d, P): DiffFunElt, PlcFunElt → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-valuation-difffunelt-plcfunelt)
- [`Divisor(d): DiffFunElt → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-divisor-difffunelt)
- [`Residue(d, P): DiffFunElt, PlcFunElt → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-residue-difffunelt-plcfunelt)
- [Example: Functions on Differentials](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#example-ex-7c77de)
- [`Module(L, R): SeqEnum[ DiffFunElt ], Rng → Mod, Map, SeqEnum[ ModElt ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-module-seqenum-difffunelt-rng)
- [`Relations(L, R): SeqEnum[ DiffFunElt ], Rng → ModTupRng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-relations-seqenum-difffunelt-rng)
- [`Relations(L, R, m): SeqEnum[ DiffFunElt ], Rng, RngIntElt → ModTupRng`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-relations-seqenum-difffunelt-rng-rngintelt)
- [Example: Differentials as a Module](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#example-ex-3e031a)
- [`Cartier(b): DiffFunElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-cartier-difffunelt)
- [`Cartier(b, r): DiffFunElt, RngIntElt → DiffFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-cartier-difffunelt-rngintelt)
- [Other](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#other)
- [`CartierRepresentation(F): FldFunG → AlgMatElt, SeqEnum[DiffFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-cartierrepresentation-fldfung)
- [`CartierRepresentation(F, r): FldFunG, RngIntElt → AlgMatElt, SeqEnum[DiffFunElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#function-cartierrepresentation-fldfung-rngintelt)
- [Example: Trivial Cartier Action on Differentials](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/differentials.md#example-ex-78c02f)
- [Weil Descent](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/weil_descent.md)
- [`WeilDescent(E,k): FldFun, FldFin → FldFunG, Map`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/weil_descent.md#function-fldfung-weil-desc)
- [`ArtinSchreierExtension(c,a,b): FldFin, FldFin, FldFin → FldFun`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/weil_descent.md#function-artinschreierextension-fldfin-fldfin-fldfin)
- [`WeilDescentDegree(E,k): FldFun, FldFin → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/weil_descent.md#function-weildescentdegree-fldfun-fldfin)
- [`WeilDescentGenus(E,k): FldFun, FldFin → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/weil_descent.md#function-weildescentgenus-fldfun-fldfin)
- [`MultiplyFrobenius(b,f,F): RngElt, RngUPolElt, Map → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/weil_descent.md#function-multiplyfrobenius-rngelt-rngupolelt-map)
- [Example: GHS Weil Descent](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/weil_descent.md#example-ex-469a62)
- [Function Field Database](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md)
- [Creation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#creation)
- [`FunctionFieldDatabase(q, d): RngIntElt, RngIntElt → DB`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#function-functionfielddatabase-rngintelt-rngintelt)
- [`sub< D | : parameters>: DB → DB`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#constructor-alffdb-sub)
- [Access](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#access)
- [`BaseField(D): DB → FldFin`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#function-basefield-db)
- [`CoefficientField(D): DB → FldFin`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#function-coefficientfield-db)
- [`Degree(D): DB → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#function-degree-db)
- [`# D: DB → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#operation-operation-db-rngintelt)
- [`NumberOfFields(D): DB → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#function-numberoffields-db)
- [`FunctionFields(D): DB → [ FldFunG ]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#function-functionfields-db)
- [Example: Genus Distribution in the Function Field Database](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/database.md#example-ex-f1c514)
- [The Montes Algorithm](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md)
- [`Montes(f, p): RngUPolElt, RngUPolElt → SeqEnum, SeqEnum, RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#function-montes-rngupolelt-rngupolelt)
- [Example: Montes Algorithm for a Polynomial](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#example-ex-531fa2)
- [`Montes(K, p): FldArith, RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#procedure-montes-fldarith-rngelt)
- [Example: Montes Algorithm for a Function Field](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#example-ex-bec397)
- [`SFL(P, s): OMIdl, RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#procedure-sfl-omidl-rngintelt)
- [Example: Refining an OM Prime Ideal Approximation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#example-ex-8aed80)
- [`SetUseMontes(f): BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#procedure-setusemontes-boolelt)
- [`SetUseMontes(t, f): Cat, BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#procedure-setusemontes-cat-boolelt)
- [`GetUseMontes(t): Cat → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#function-getusemontes-cat)
- [`SetVerbose("Montes", v): MonStgElt, RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#procedure-setverbose-monstgelt-rngintelt-montes)
- [Ideals in OM Representation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md)
- [`Ideal(I): OMIdl → RngOrdIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-ideal-omidl)
- [`OMRepresentation(I): RngFunOrdIdl → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-omrepresentation-rngfunordidl)
- [`OMRepresentation(I): RngOrdFracIdl → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-omrepresentation-rngordfracidl)
- [`OMRepresentation(L, S): FldArith, [FldArithElt] → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-omrepresentation-fldarith-fldarithelt)
- [`OMRepresentation(L, a): FldArith, FldArithElt → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-omrepresentation-fldarith-fldarithelt-2)
- [`OMRepresentation(L, a): FldArith, RngElt → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-omrepresentation-fldarith-rngelt)
- [Example: Converting OM Prime Ideals to Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#example-ex-c49fbd)
- [Ideal Arithmetic](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#ideal-arithmetic)
- [`I + J: OMIdl, OMIdl → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#operation-op-plus-omidl-omidl)
- [`I * J: OMIdl, OMIdl → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#operation-op-times-omidl-omidl)
- [`I / J: OMIdl, OMIdl → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#operation-op-div-omidl-omidl)
- [`I ^ n: OMIdl, RngIntElt → OMIdl`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#operation-op-pow-omidl-rngintelt)
- [Example: Arithmetic of OM Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#example-ex-0a493e)
- [Ideal Predicates](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#ideal-predicates)
- [`IsOne(I): OMIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-isone-omidl)
- [`IsZero(I): OMIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-iszero-omidl)
- [`I eq J: OMIdl, OMIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#operation-op-eq-omidl-omidl)
- [`a in I: RngElt, OMIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#operation-op-in-rngelt-omidl)
- [`I subset J: OMIdl, OMIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#operation-op-subset-omidl-omidl)
- [`IsPrime(I): OMIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-isprime-omidl)
- [`IsIntegral(I): OMIdl → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-rep.md#function-isintegral-omidl)
- [Ideal Operations](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md)
- [`pIntegralBasis(I, p): OMIdl, RngElt → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-pintegralbasis-omidl-rngelt)
- [`SIntegralBasis(I, S): OMIdl, SeqEnum → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-sintegralbasis-omidl-seqenum)
- [`SIntegralBasis(I, S): RngOrdFracIdl, [RngIntElt] → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-sintegralbasis-rngordfracidl-rngintelt)
- [`SIntegralBasis(I, S): RngFunOrdIdl, [RngUPolElt] → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-sintegralbasis-rngfunordidl-rngupolelt)
- [`Basis(I): OMIdl → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-basis-omidl)
- [Example: Integral Basis of an OM Ideal](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#example-ex-70e335)
- [`TwoElement(I): OMIdl → FldArithElt, FldArithElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-twoelement-omidl)
- [`Norm(I): OMIdl → RngElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-norm-omidl)
- [`Valuation(alpha, P : parameters): FldArithElt, OMIdl → RngIntElt, FldElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-valuation-fldarithelt-omidl)
- [`Valuation(alpha, P : parameters): FldRatElt, OMIdl → RngIntElt, FldElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-valuation-fldratelt-omidl)
- [`Valuation(alpha, P : parameters): RngIntElt, OMIdl → RngIntElt, FldElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-valuation-rngintelt-omidl)
- [`Valuation(alpha, P : parameters): RngUPolElt, OMIdl → RngIntElt, FldElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-valuation-rngupolelt-omidl)
- [`Valuation(I, P): OMIdl, OMIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-valuation-omidl-omidl)
- [`a mod P: FldArithElt, OMIdl → FldArithElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#operation-operation-mod-fldarithelt-omidl-fldarithelt)
- [`Reduction(a, P): FldArithElt, OMIdl → FldArithElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-reduction-fldarithelt-omidl)
- [`Reduction(a, P, m): FldArithElt, OMIdl, RngIntElt → [FldArithElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-reduction-fldarithelt-omidl-rngintelt)
- [`Factorization(I): OMIdl → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-factorization-omidl)
- [`Factorisation(I): OMIdl → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-factorisation-omidl)
- [Example: Creating an OM Ideal](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#example-ex-fbb1be)
- [`ResidueField(I): OMIdl → Fld`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-residuefield-omidl)
- [`Degree(I): OMIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#function-degree-omidl)
- [Example: Degree and Residue Field of OM Ideals](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/ideal-op.md#example-ex-7d7ba3)
- [Divisors in OM representation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md)
- [Constructing Divisors in OM representation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#constructing-divisors-in-om-representation)
- [`OMDivisor(If, Ii): OMIdl, OMIdl → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-omdivisor-omidl-omidl)
- [`OMDivisor(I): OMIdl → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-omdivisor-omidl)
- [`OMDivisor(z): FldFunElt → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-omdivisor-fldfunelt)
- [`PoleDivisor(a): FldFunElt → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-poledivisor-fldfunelt)
- [`ZeroDivisor(a): FldFunElt → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-zerodivisor-fldfunelt)
- [`OMDivisorOfDegreeOne(F): FldFun → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-omdivisorofdegreeone-fldfun)
- [`ReferenceDivisor(F): FldFun → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-referencedivisor-fldfun)
- [`OMDivisor(D): DivFunElt → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-omdivisor-divfunelt)
- [Example: Constructing Divisors in OM Representation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#example-ex-42c816)
- [Arithmetic with Divisors in OM representation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#arithmetic-with-divisors-in-om-representation)
- [`k * D: RngIntElt, OMDiv → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#operation-op-times-rngintelt-omdiv)
- [`D1 + D2: OMDiv, OMDiv → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#operation-op-plus-omdiv-omdiv)
- [`D1 - D2: OMDiv, OMDiv → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#operation-op-minus-omdiv-omdiv)
- [`GCD(D1, D2): OMDiv, OMDiv → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-gcd-omdiv-omdiv)
- [`Gcd(D1, D2): OMDiv, OMDiv → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-gcd-omdiv-omdiv-2)
- [`GreatestCommonDivisor(D1, D2): OMDiv, OMDiv → OMDiv`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-greatestcommondivisor-omdiv-omdiv)
- [Example: Arithmetic of OM Divisors](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#example-ex-31985e)
- [Predicates on Divisors in OM representation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#predicates-on-divisors-in-om-representation)
- [`IsEffective(D): OMDiv → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-iseffective-omdiv)
- [`IsPositive(D): OMDiv → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-ispositive-omdiv)
- [`IsPrincipal(D): OMDiv → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-isprincipal-omdiv)
- [`D1 eq D2: OMDiv, OMDiv → BoolElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#operation-op-eq-omdiv-omdiv)
- [Example: Predicates on OM Divisors](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#example-ex-c8e530)
- [Other Operations on Divisors in OM representation](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#other-operations-on-divisors-in-om-representation)
- [`Divisor(D): OMDiv → DivFunElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-divisor-omdiv)
- [`Valuation(D, P): OMDiv, OMIdl → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-valuation-omdiv-omidl)
- [`Support(D): OMDiv → [OMIdl], [RngIntElt]`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-support-omdiv)
- [`Height(D): OMDiv → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-height-omdiv)
- [`CanonicalHeight(D): OMDiv → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-canonicalheight-omdiv)
- [`Degree(D): OMDiv → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-degree-omdiv)
- [`Basis(D): OMDiv → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-basis-omdiv)
- [`Dimension(D): OMDiv → RngIntElt`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-dimension-omdiv)
- [`ReducedBasis(D): OMDiv → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-reducedbasis-omdiv)
- [`SemiReducedBasis(D): OMDiv → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-semireducedbasis-omdiv)
- [`SuccessiveMinima(D): OMDiv → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-successiveminima-omdiv)
- [`ApproximatedSuccessiveMinima(D): OMDiv → SeqEnum`](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#function-approximatedsuccessiveminima-omdiv)
- [Example: Operations on OM Divisors](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/divisors-in-om-representation.md#example-ex-e8bf40)
- [Bibliography](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/bibliography.md)
### Introduction
This version of Magma is based on the KANT Version 4 system for algebraic function field computations. Special functions for rational function fields are described in Chapter [Rational Function Fields](https://docs.magma-maths.org/GlobalFields/RationalFunctionFields/index-rational-function-fields.md#fldfunrat-main).
An algebraic function field $F/k$ (in one variable) over a field $k$ is a field extension $F$ of $k$ such that $F$ is a finite field extension of $k(x)$ for an element $x \in F$ which is transcendental over $k$. As a Magma object it is of type `FldFun` with elements of type `FldFunElt`. For perfect $k$ it is always possible to choose $x \in F$ so that $F/k(x)$ is also separable. For such $x$ there exists a primitive element $\alpha \in F$ with $F = k(x, \alpha)$ where $\alpha$ is a root of an irreducible, separable polynomial in $k(x)[y]$.
#### Representations of Fields
All function fields in Magma can be represented as described above, i.e. as $k(x, \alpha)$ where $x$ is transcendental and $\alpha$ is algebraic. This is the representation which has the most functionality.
Alternatively, one may wish to consider a function field as a combined transcendental and algebraic extension of its constant field, more like a curve. Such a field would be a quotient of $k[x, y]$. Fields in this representation do not have orders.
Algebraic function fields may be extended to create relative finite extensions of $k(x)$ like $F = k(x, \alpha_1, \ldots, \alpha_n)$. The functionality described in this chapter which is not available for these relative extensions is that involving series rings, galois groups, and subfields.
It is also possible to make non–simple extensions where more than one root of a polynomial is added at each step by extending by several polynomials. These extensions have the same functionality as the relative finite extensions, except that primitive elements and some functions involving differentials are not available.
Function fields represented as finite extensions may have orders. Some orders will have a basis different to that of their function fields. Orders may have a field of fractions. A field of fractions of an order is isomorphic to the function field of the order, however its elements are represented with respect to the basis of the order. So there are 3 different types of rings covered in this chapter, function fields (`FldFun`), orders of function fields (`RngFunOrd`) and fields of fractions of orders of function fields (`FldFunOrd`). Each ring type has its own corresponding element type.
### Creation of Algebraic Function Fields and their Orders
#### Creation of Algebraic Function Fields
##### `ext< K | f >: FldFunRat, RngUPolElt -> FldFun`
##### `ext< K | f >: FldFun, RngUPolElt -> FldFun`
##### `FunctionField(f : parameters): RngUPolElt -> FldFun`
```magma
Check : BoolElt Default: true
Global: BoolElt Default: true
```
Let $k$ be a field and $K = k(x)$ or $K = k(x, \alpha_1, \ldots, \alpha_r)$ some finite extension of $k(x)$. Given an irreducible and separable polynomial $f \in K[y]$ of degree greater than zero with coefficients within $K$, create the algebraic function field $F= K[y] / \langle f \rangle = k(x, \alpha_1, \ldots, \alpha_r, \alpha)$ obtained by adjoining a root $\alpha$ of $f$ to $K$. $F$ will be viewed as a (finite) extension of $K$. The polynomial $f$ is also allowed to be $\in k[x][y]$.
The optional parameter `Check` may be used to prevent some conditions from being tested. The default is `Check := true`, so that $f$ is verified to be irreducible and separable. The optional parameter `Global` may be used to allow another copy of the field to be returned if it is set to `false`, otherwise if a field has already been constructed using $f$ over $K$ and has not been deleted then the existing field will be returned.
The angle bracket notation may be used to assign the root $\alpha$ to an identifier: `F := FunctionField(f)`.
##### `FunctionField(f : parameters): RngMPolElt -> FldFun`
```magma
Check : BoolElt Default: true
Global: BoolElt Default: true
```
Let $k$ be a field. Given an irreducible polynomial $f \in k[x, y]$ of degree greater than zero, create the algebraic function field $F$ which is the field of fractions of $k[x, y] / \langle f \rangle$. The polynomial $f$ must be separable in at least one variable. $F$ will be viewed as (infinite) extension of $k$.
The optional parameter `Check` may be used to prevent some conditions from being tested. The default is `Check := true`, so that $f$ is verified to be irreducible and separable in at least one variable. The optional parameter `Global` may be used to allow another copy of the field to be returned if it is set to `false`, otherwise if a field has already been constructed using $f$ over $K$ and has not been deleted then the existing field will be returned.
The angle bracket notation may be used to assign the images of $x,y$ in $F$ to identifiers: `F := FunctionField(f)`.
##### `FunctionField(S): [RngUPolElt] -> FldFun`
```magma
Check: BoolElt Default: true
```
Return the function field $F$ whose defining polynomials are the polynomials in the sequence $S$. If `Check` is set to `false` then it will not be checked the polynomials actually define a field.
##### `HermitianFunctionField(p, d): RngIntElt, RngIntElt -> FldFun`
##### `HermitianFunctionField(q): RngIntElt -> FldFun`
Create the Hermitian function field $F = {\bf F}_{q^2}(x, \alpha)$ defined by $\alpha^q + \alpha = x^{q+1}$, where $q$ is the $d$-th power of the prime number $p$.
##### `sub: FldFun, [] -> FldFun`
##### `sub: FldFun, [] -> FldFun`
The subfield of the function field $F$ containing the elements in the sequence $S$ or the elements $s_i$.
##### `AssignNames(~F, s): FldFun, [ MonStgElt ]`
##### `AssignNames(~a, s): FldFunElt, [ MonStgElt ]`
Procedure to change the name of the generating element(s) in the function field $F$ ($a$ in $F$) to the contents of the sequence of strings $s$, which must have length $1$ or $2$ in this case.
This procedure only changes the name(s) used in printing the elements of $F$. It does *not* assign to any identifier(s) the value(s) of the generator(s) in $F$; to do this, use an assignment statement, or use angle brackets when creating the field.
Note that since this is a procedure that modifies $F$, it is necessary to have a reference `~F` to $F$ (or $a$) in the call to this function.
##### `FunctionField(R): Rng -> FldFunG`
```magma
Global: BoolElt Default: true
Type : Cat Default: FldFunRat
```
Return the rational function field over $R$ in one variable. If `Global` is `false` then create a new copy of the field, otherwise reuse any globally created field which already exists. If `Type` is `FldFun` create the field as an algebraic function field, otherwise create as a rational function field.
##### `Example: Creating a Function Field Extension (ex-4f7d5b)`
Let ${\bf F}_{5}$ be the finite field of five elements. To create the function field extension ${\bf F}_{5}(x, \alpha) / {\bf F}_{5}(x)$, where $\alpha$ satisfies $\alpha^2 = 1/x,$ one may proceed in the following, equivalent ways:
```magma
> R := FunctionField(GF(5));
> P := PolynomialRing(R);
> F := FunctionField(y^2 - 1/x);
> F;
Algebraic function field defined over Univariate rational function field over
GF(5)
Variables: x by
y^2 + 4/x
```
or
```magma
> R := PolynomialRing(GF(5));
> P := PolynomialRing(R);
> F := FunctionField(x*y^2 - 1);
> F;
Algebraic function field defined over Univariate rational function field over
GF(5)
Variables: x by
x*y^2 + 4
```
or
```magma
> R := FunctionField(GF(5));
> P := PolynomialRing(R);
> F := ext< R | y^2 - 1/x >;
> F;
Algebraic function field defined over Univariate rational function field over
GF(5)
Variables: x by
y^2 + 4/x
```
##### `Example: Creating a Relative Extension (ex-b33520)`
An extension of $F$ may be created as follows.
```magma
> R := PolynomialRing(F);
> FF := FunctionField(y^3 - x/alpha : Check := false);
> FF;
Algebraic function field defined over F by
y^3 + 4*x^2*alpha
```
##### `Example: Creating a Non-Simple Extension (ex-5f689c)`
To create a non–simple extension:
```magma
> R := FunctionField(GF(5));
> P := PolynomialRing(R);
> FF := FunctionField([y^2 - 1/x, y^3 + x]);
> FF;
Algebraic function field defined over Univariate rational function field over
GF(5) by
y^2 + 4/x
y^3 + x
```
or
```magma
> P := PolynomialRing(F);
> FF := FunctionField([y^2 - x/alpha, y^3 + x]);
> FF;
Algebraic function field defined over F by
y^2 + 4*x^2*alpha
y^3 + x
```
##### `Example: Hermitian Function Field (ex-5c6691)`
The creation of an Hermitian function field:
```magma
> F := HermitianFunctionField(9);
> F;
Algebraic function field defined over GF(3^4) by
y^9 + y + 2*x^10
```
#### Construction of Orders of Algebraic Function Fields
Equation orders, maximal orders, and other orders of algebraic function fields can be constructed.
For more information about the Montes algorithm which can be used to compute maximal orders, see Section [The Montes Algorithm](https://docs.magma-maths.org/GlobalFields/AlgebraicFunctionFields/montes.md#montes-fld-fun).
##### `EquationOrderFinite(F): FldFun -> RngFunOrd`
Create the ‘finite’ equation order of the function field $F/k(x, \alpha_1, \ldots, \alpha_r)$, i.e. $k[x, d_1 \alpha_1, \ldots, d_r \alpha_r, d \alpha]$ where $d_j, d \in k[x]$ is chosen such that $d_j \alpha_j, d \alpha$ are integral over $k[x]$.
##### `MaximalOrderFinite(F): FldFun -> RngFunOrd`
```magma
Al: MonStgElt Default:
```
Construct the ‘finite’ maximal order of the function field $F/k(x, \alpha_1, \ldots, \alpha_r)$. This is the integral closure of $k[x, d_1 \alpha_1, \ldots, d_r \alpha_r]$ in $F$. If $F$ is an extension of a rational function field over ${\mathbb{Q}}$ or ${\mathbb{F}}_q$ by a single monic integral polynomial and the parameter `Al` is set to `"Montes"` then the Montes algorithm [[Sta18](https://docs.magma-maths.org/references.md#cite-2015arxiv150601904s)] will be used for the computation.
##### `EquationOrderInfinite(F): FldFun -> RngFunOrd`
Create the ‘infinite’ equation order of the function field $F/k(x, \alpha_1, \ldots, \alpha_r)$, i.e. $o_{\infty}[\alpha_1, \ldots, \alpha_r, \beta]$ where $o_{\infty}$ denotes the valuation ring of the degree valuation in $k(x)$ and $\beta$ is a primitive element of $F/k(x, \alpha_1, \ldots, \alpha_r)$ which is integral over $o_{\infty}$.
##### `MaximalOrderInfinite(F): FldFun -> RngFunOrd`
Create the ‘infinite’ maximal order of the function field $F/k(x, \alpha_1, \ldots, \alpha_r)$. This is the integral closure of $o_{\infty}$ in $F$.
##### `IntegralClosure(R, F): Rng, FldFun -> RngFunOrd`
The integral closure of the subring $R$ of the function field $F$ in itself.
##### `EquationOrder(O): RngFunOrd -> RngFunOrd`
The equation order of the order $O$. An order whose basis is a transformation of that of $O$ and is a power basis.
##### `MaximalOrder(O): RngFunOrd -> RngFunOrd`
```magma
Discriminant: Any Default:
Ramification: SeqEnum Default:
Al : MonStgElt Default: "Auto"
MaximalOrder: Verbose Default: 5
```
The maximal order of the order $O$ of an algebraic function field.
If $O$ is a radical (pure) extension then specific code is used to calculate each $p$-maximal order, rather than the Round $2$ method. In this case we can compute a pseudo basis for the $p$-maximal orders knowing only the valuation of the constant coefficient of the defining polynomial at $p$ [[Sut12](https://docs.magma-maths.org/references.md#cite-suth-max-kummer)].
If $O$ is an Artin–Schreier extension then the maximal order can be computed directly without computing the $p$-maximal orders. The proof of Proposition III.7.8 of [[Sti93](https://docs.magma-maths.org/references.md#cite-stichtenoth)] gives us a start on some elements which are a basis for the maximal order [[Sut13](https://docs.magma-maths.org/references.md#cite-suth-artin)].
If the `Discriminant` or `Ramification` parameters are supplied an algorithm ([[BL94c](https://docs.magma-maths.org/references.md#cite-buchman-lenstra)], Theorems 1.2 and 7.6) which can compute the maximal order given the discriminant of the maximal order will be used. `Discriminant` must be an element of the coefficient ring of $O$ if $O$ is a non relative order and must be an ideal of $O$ if $O$ is a relative order. `Ramification` must contain elements of the coefficient ring if $O$ is a non relative order and must contain ideals of $O$ if $O$ is a relative order. The ramification sequence is taken to contain prime factors of the discriminant. Only one of these parameters can be specified and if one of them is then `Al` cannot be specified.
If $O$ is an extension of a polynomial ring over ${\mathbb{Q}}$ or ${\mathbb{F}}_q$ by a single monic integral polynomial and the parameter `Al` is set to `"Montes"` then the Montes algorithm [[Sta18](https://docs.magma-maths.org/references.md#cite-2015arxiv150601904s)] will be used to compute the maximal order. Otherwise `Al` may be set to `"Round2"` to avoid using the algorithms for the special cases above.
##### `SetOrderMaximal(O, b): RngFunOrd, BoolElt`
Set the order $O$ of a function field to be maximal if $b$ is `true` and to be non–maximal if $b$ is `false`.
##### `ext: RngFunOrd, RngUPolElt -> RngFunOrd`
```magma
Check: Bool Default: true
```
The order $O$ with a root of $f$ adjoined.
##### `Example: Creating Orders of a Function Field (ex-68faea)`
Creation of orders is shown below.
```magma
> PR := PolynomialRing(Rationals());
> P := PolynomialRing(PR);
> FR1 := FunctionField(y^3 - x*y^2 + y + x^4);
> P