Related Structures#

Parent and Category#

Function fields form the Magma category FldFun and function field orders form the Magma category RngFunOrd. The notional power structures exist as parents of function fields and their orders but allow no operations.

Category(F): FldFun -> Cat#

Category(O): RngFunOrd -> Cat#

Parent(F): FldFun -> Pow#

Parent(O): RngFunOrd -> Pow#

Other Related Structures#

More interesting related structures (than above) are listed below.

PrimeRing(F): FldFun -> Rng#
PrimeField(F): FldFun -> Rng#
PrimeRing(O): RngFunOrd -> Rng#: The prime field of the function field \(F\) or the order \(O\) (prime ring of the constant field).

ConstantField(F): FldFunG -> Rng#
DefiningConstantField(F): FldFunG -> Rng#: The constant field \(k\), where \(F = k(x, \alpha)\).

ExactConstantField(F): FldFunG -> Rng, Map#: The exact constant field of the algebraic function field \(F/k\), i.e. the algebraic closure in \(F\) of the constant field \(k\) of \(F\), together with the inclusion map.

BaseRing(F): FldFun -> Rng#
BaseField(F): FldFun -> Rng#
CoefficientRing(F): FldFun -> Rng#
CoefficientField(F): FldFun -> Rng#: The rational function field \(k(x)\) if the function field \(F\) is an extension of \(k(x)\) and \(k\) if \(F\) is an extension of \(k\). If \(F\) is an extension of another algebraic function field then this field will be returned.

ISABaseField(F, G): Fld, Fld -> BoolElt#: Applies to more general fields within Magma than function fields. Returns whether \(G\) is amongst the recursively defined base fields of \(F\).

BaseRing(O): RngFunOrd -> Rng#
CoefficientRing(O): RngFunOrd -> Rng#: The polynomial algebra \(k[x]\) if the order \(O\) is finite or the degree valuation ring if \(O\) is infinite. If \(O\) is an extension of another order of an algebraic function field this order will be returned.

BaseRing(FF): FldFunOrd -> Rng#
BaseField(FF): FldFunOrd -> Rng#
CoefficientRing(FF): FldFunOrd -> Rng#
CoefficientField(FF): FldFunOrd -> Rng#: Given a field of fractions \(FF\) of an order \(O\) return the field of fractions of the coefficient ring of \(O\).

SubOrder(O): RngFunOrd -> RngFunOrd#: For a non equation order \(O\) returns the order which \(O\) was created as a transformation of. This order is one transformation closer to the equation order.

FunctionField(O): RngFunOrd -> FldFun#: The function field which \(O\) is an order of.

FieldOfFractions(O): RngFunOrd -> FldFunOrd#
FieldOfFractions(FF): FldFunOrd -> FldFunOrd#
FieldOfFractions(F): FldFun -> FldFun#: Given an order \(O\), this function returns the field of fractions, a field with the same basis as \(O\). On a function field or a field of fractions this function is trivial.

Order(FF): FldFunOrd -> RngFunOrd#: Given a field of fractions \(FF\) return the order \(O\) which is the ring of integers of \(FF\).

RationalExtensionRepresentation(F): FldFunG -> FldFun#: The function field \(F\) represented as an extension of a rational function field. This function gives the representation of function fields \(F/k\) as finite extensions.

AbsoluteOrder(O): RngFunOrd -> RngFunOrd#: The order \(O\) as an extension of its bottom coefficient ring, (i.e. the order of the RationalExtensionRepresentation of the field of fractions of \(O\) corresponding to \(O\)).

AbsoluteFunctionField(F): FldFunG -> FldFunG#: The function field \(F\) expressed as an extension of its constant field.

UnderlyingRing(F): FldFunG -> FldFunG#
UnderlyingField(F): FldFunG -> FldFunG#
UnderlyingRing(F, R): FldFunG, Rng -> FldFunG#
UnderlyingField(F, R): FldFunG, Rng -> FldFunG#: Return the underlying ring of the function field \(F\) over \(R\). This is \(F\) expressed as an extension of \(R\). If \(R\) is not given then it is taken to be the coefficient field of the coefficient field of \(F\). The field \(R\) must appear in the tower of coefficient fields under \(F\).

Embed(F, L, a): FldFun, FldFun, FldFunElt#
Embed(F, L, s): FldFun, FldFun, [FldFunElt]#: Install the embedding of \(F\) into \(L\) with the image(s) of the primitive element(s) of \(F\) being the element \(a\) in \(L\) or the images in \(s\) in \(L\).

HasEmbedding(O1, O2): RngFunOrd, RngFunOrd -> Bool#

Print: RngIntElt                    Default: 0

Return whether there is an embedding of \(O1\) into \(O2\) already installed. For information about the installed embedding set the Print parameter \(> 0\) with higher values providing more information than lower.

CanComputeEmbedding(O1, O2): RngFunOrd, RngFunOrd -> Bool#

Print: RngIntElt                    Default: 0

Return whether an embedding of \(O1\) into \(O2\) can be computed. For information about the installed embedding set the Print parameter \(> 0\) with higher values providing more information than lower.

Places(F): FldFunG -> PlcFun#: The set of places of the algebraic function field \(F/k\).

DivisorGroup(F): FldFun -> DivFun#: The group of divisors of the algebraic function field \(F/k\).

DifferentialSpace(F): FldFun -> DiffFun#: The space of differentials of the algebraic function field \(F/k\).

Example: Related Structures (ex-a0477f)#

> R<x> := FunctionField(GF(5));
> P<y> := PolynomialRing(R);
> f := y^3 + (4*x^3 + 4*x^2 + 2*x + 2)*y^2 + (3*x + 3)*y + 2;
> F<alpha> := FunctionField(f);
> ConstantField(F);
Finite field of size 5
> CoefficientField(F);
Univariate rational function field over GF(5)
Variables: x
> CoefficientRing(MaximalOrderFinite(F));
Univariate Polynomial Ring in x over GF(5)
> FieldOfFractions(IntegralClosure(ValuationRing(R), F));
Algebraic function field defined over Univariate rational function field over
GF(5)
Variables: x by
y^3 + (4*x^3 + 4*x^2 + 2*x + 2)*y^2 + (3*x + 3)*y + 2
> Order(IntegralClosure(ValuationRing(R), F),
>     MatrixAlgebra(CoefficientRing(MaximalOrderInfinite(F)), 3)!4,
>     CoefficientRing(MaximalOrderInfinite(F))!1);
Maximal Order of F over Valuation ring of Univariate rational function field
over GF(5) with generator 1/x
> SubOrder($1);
Maximal Order of F over Valuation ring of Univariate rational function field
over GF(5) with generator 1/x
> Places(F);
Set of places of F
> DivisorGroup(F);
Divisor group of F

Run in calculator

Example: Related Structures Rat Ext (ex-efbad3)#

Output from UnderlyingRing is shown.

> PF<x> := PolynomialRing(GF(31, 3));
> P<y> := PolynomialRing(PF);
> FF1<b> := ext<FieldOfFractions(PF) | y^2 - x^3 + 1>;
> P<y> := PolynomialRing(FF1);
> FF2<d> := ext<FF1 | y^3 - b*x*y - 1>;
> RationalExtensionRepresentation(FF2);
Algebraic function field defined over Univariate rational function field over
GF(31^3) by
y^6 + 29*y^3 + (30*x^5 + x^2)*y^2 + 1
> UnderlyingRing(FF2);
Algebraic function field defined over Univariate rational function field over
GF(31^3) by
y^6 + 29*y^3 + (30*x^5 + x^2)*y^2 + 1
> UnderlyingRing(FF2, FieldOfFractions(PF));
Algebraic function field defined over GF(31^3) by
$.1^6 + 29*$.1^3 + 30*$.1^2*$.2^5 + $.1^2*$.2^2 + 1

Run in calculator

WeilRestriction(E, n): FldFun, RngIntElt -> FldFun, UserProgram#

Reduction: BoolElt                    Default: true
verbose  : WeilRes                    Default: Verbose : 1

A hyperelliptic function field in the Weil restriction over \({\bf F}_{q}\) of the elliptic function field \(E\): \(y^2 + xy + x^3 + ax^2 + b\) defined over \({\bf F}_{q^n}\) where \(q\) is a power of \(2\). Also returns a function which can be used to map a place (not a pole or zero of \(x\)) of \(F\) into a divisor of the result. See [P. Gaudry and Smart, 2000]. Reduction indicates whether a (possibly quite expensive) reduction step is performed at the end of the computation. It defaults to true.

ConstantFieldExtension(F, E): FldFun, Rng -> FldFun, Map#: Return the function field with constant field \(E\) which contains the function field \(F\). The ring \(E\) must cover the constant field of \(F\). If \(E\) is contained in the exact constant field of \(F\) then \(F\) and the new field will be isomorphic.

Example: cfe (ex-01df8e)#

Changing the constant field to the exact constant field is shown below.

> P<x> := PolynomialRing(Rationals());
> P<y> := PolynomialRing(P);
> F<c> := FunctionField(y^6 + y + 2);
> E<a> := ExactConstantField(F);
> C, r := ConstantFieldExtension(F, E);
> r(c);
1/16*(a^5 + 4*a^4 + 6*a^3 + 4*a^2 + a)
> $1 @@ r;
c
> e := Random(C, 2);
> e @@ r;
1/2*x*c^5 - 3*x*c^4 + (-12*x + 8)*c^3 + (-16*x + 24)*c^2 + (-8*x + 16)*c - 1
> r($1);
1/2*(-a^5 - a^2 + a)*$.1 + a^5 + a^4 - a^3 - a^2 - 1

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MonicModel(F): FldFun -> FldFun#: Given an algebraic function field \(F\) return a function field which is isomorphic to \(F\) and defined by a monic polynomial.

Reduce(O): RngFunOrd -> RngFunOrd#: Given an order \(O\) belonging to a function field \(F\), this function returns the order obtained by applying size-reduction to the basis of \(O\).

Localization(O, p): RngFunOrd, RngFunOrdIdl -> RngVal, Map#: Given an order \(O\) of an algebraic function field and a prime ideal \(p\) of \(O\), return the localization of \(O\) at \(p\) and the map from \(O\) into the localization.