Element Operations

See also Section Generic Element Functions.

Arithmetic Operators

+ a: FldFinElt -> FldFinElt
- a: FldFinElt -> FldFinElt
a + b: FldFinElt, FldFinElt -> FldFinElt
a - b: FldFinElt, FldFinElt -> FldFinElt
a * b: FldFinElt, FldFinElt -> FldFinElt
a / b: FldFinElt, FldFinElt -> FldFinElt
a ^ k: FldFinElt, RngIntElt -> FldFinElt
a +:= b: FldFinElt, FldFinElt -> FldFinElt
a -:= b: FldFinElt, FldFinElt -> FldFinElt
a *:= b: FldFinElt, FldFinElt -> FldFinElt

Equality and Membership

a eq b: FldFinElt, FldFinElt -> BoolElt
a ne b: FldFinElt, FldFinElt -> BoolElt
a in F: FldFinElt, Rng -> BoolElt
a notin F: FldFinElt, Rng -> BoolElt

Parent and Category

Parent(a): FldFinElt -> FldFin
Category(a): FldFinElt -> Cat

Predicates on Ring Elements

IsZero(a): FldFinElt -> BoolElt
IsOne(a): FldFinElt -> BoolElt
IsMinusOne(a): FldFinElt -> BoolElt
IsNilpotent(a): FldFinElt -> BoolElt
IsIdempotent(a): FldFinElt -> BoolElt
IsUnit(a): FldFinElt -> BoolElt
IsZeroDivisor(a): FldFinElt -> BoolElt
IsRegular(a): FldFin -> BoolElt
IsIrreducible(a): FldFinElt -> BoolElt
IsPrime(a): FldFinElt -> BoolElt
IsPrimitive(a): FldFinElt -> BoolElt

Returns true if and only if the element \(a\) of \(F\) is a primitive element for \(F\) (i.e., if and only if the multiplicative order of \(a\) is \(\#F-1\)).

IsPrimitive(f): RngUPolElt -> BoolElt

Given a univariate polynomial \(f\in F[x]\), over a finite field \(F\), such that the degree of \(f\) is greater than or equal to 1, this function returns true if and only if \(f\) defines a primitive extension \(G=F[x]/f\) of \(F\) (that is, \(x\) is primitive in \(G\)).

IsNormal(a): FldFinElt -> BoolElt

Returns true if and only if the element \(a\) of \(F\) generates a normal basis for the field over the ground field, that is, if and only if \(a, a^q, \ldots, a^{q^{n-1}}\) form a basis for \(F\) over the ground field \(G={\bf F}_{q}\).

IsNormal(a, E): FldFinElt -> BoolElt

Returns true if and only if the element \(a\) of the finite field \(F\) with \(q^n\) elements generates a normal basis for \(F\) over its subfield \(E\), that is, if and only if \(a, a^q, \ldots, a^{q^{n-1}}\) form a basis for \(F\) over \(E\) for \(q=\#E\).

IsSquare(a): FldFinElt -> BoolElt

Given a finite field element \(a\in F\), this function returns either true and an element \(b\in F\) such that \(b^2=a\), or it returns false in the case that such an element does not exist.

Minimal and Characteristic Polynomial

MinimalPolynomial(a): FldFinElt -> RngUPolElt

The minimal polynomial of the element \(a\) of the field \(F\), relative to the ground field of \(F\). This is the unique minimal-degree monic polynomial with coefficients in the ground field, having \(a\) as a root.

MinimalPolynomial(a, E): FldFinElt, FldFin -> RngUPolElt

The minimal polynomial of the element \(a\) of the field \(F\), relative to the subfield \(E\) of \(F\). This is the unique minimal-degree monic polynomial with coefficients in \(E\), having \(a\) as a root.

CharacteristicPolynomial(a): FldFinElt -> RngUPolElt

Given an element \(a\) of a finite field \(F\), return the characteristic polynomial of \(a\) with respect to the ground field of \(F\). (This polynomial is the characteristic polynomial of the companion matrix of \(a\) written as a polynomial over the ground field, and is a power of the minimal polynomial.)

CharacteristicPolynomial(a, E): FldFinElt, FldFin -> RngUPolElt

Given an element \(a\) of a finite field \(F\), return the characteristic polynomial of \(a\) with respect to the subfield \(E\) of \(F\). (This polynomial is the characteristic polynomial of the companion matrix of \(a\) written as a polynomial over \(E\), and is a power of the minimal polynomial over \(E\).)

Norm, Trace, and Frobenius

Norm(a): FldFinElt -> FldFinElt

The norm of the element \(a\) from the field \(F\) to the ground field of \(F\).

Norm(a, E): FldFinElt, FldFin -> FldFinElt

The relative norm of the element \(a\) from the field \(F\), with respect to the subfield \(E\) of \(F\). The result is an element of \(E\).

AbsoluteNorm(a): FldFinElt -> FldFinElt
NormAbs(a): FldFinElt -> FldFinElt

The absolute norm of the element \(a\), that is, the norm to the prime subfield of the parent field \(F\) of \(a\).

Trace(a): FldFinElt -> FldFinElt

The trace of the element \(a\) from the field \(F\) to the ground field of \(F\).

Trace(a, E): FldFinElt, FldFin -> FldFinElt

The relative trace of the element \(a\) from field \(F\), with respect to the subfield \(E\) of \(F\). The result is an element of \(E\).

AbsoluteTrace(a): FldFinElt -> FldFinElt
TraceAbs(a): FldFinElt -> FldFinElt

The trace of the element \(a\), that is, the trace to the prime subfield of the parent field \(F\) of \(a\).

Frobenius(a): FldFinElt -> FldFinElt

The Frobenius image of \(a\) w.r.t. the ground field of \(K\); i.e., \(a^{\#G}\), where \(G\) is the ground field of the parent of \(a\).

Frobenius(a, r): FldFinElt, RngIntElt -> FldFinElt

The \(r\)-th Frobenius image of \(a\) w.r.t. the ground field of \(K\); i.e., \(a^{(\#G)^r}\), where \(G\) is the ground field of the parent of \(a\).

Frobenius(a, E): FldFinElt, FldFin -> FldFinElt

The Frobenius image of x w.r.t. \(E\); i.e., \(x^{\#E}\).

Frobenius(a, E, r): FldFinElt, FldFin, RngIntElt -> FldFinElt

The Frobenius image of x w.r.t. \(E\); i.e., \(x^{(\#E)^r}\).

NormEquation(K, y): FldFin, FldFin -> BoolElt, FldFinElt

Given a finite field \(K\) and an element \(y\) of a subfield \(S\) of \(K\), return whether an element \(x\in K\) exists such that Norm\((x, S) = y\), and, if so, such an element \(x\) (in \(K\)).

Hilbert90(a, q): FldFinElt, RngIntElt -> FldFinElt

Given an element \(a\) of some finite field \(k\) and a power \(q\) of the characteristic of \(k\), return a solution of the Hilbert 90 equation \(x^qx^{-1}=a\). Note that the solution may be in a finite-degree extension of \(k\).

AdditiveHilbert90(a, q): FldFinElt, RngIntElt -> FldFinElt

Given an element \(a\) of some finite field \(k\) and a power \(q\) of the characteristic of \(k\), return a solution of the additive Hilbert 90 equation \(x^q - x=a\). Note that the solution may be in a finite-degree extension of \(k\).

Order and Roots

Order(a): FldFinElt -> RngIntElt

The multiplicative order of the non-zero element \(a\) of the field \(F\).

FactoredOrder(a): FldFinElt -> RngIntElt

The multiplicative order of the non-zero element \(a\) of the field \(F\) as a factorization sequence.

SquareRoot(a): FldFinElt -> FldFinElt
Sqrt(a): FldFinElt -> FldFinElt

The square root of the non-zero element \(a\) from the field \(F\), i.e., an element \(y\) of \(F\) such that \(y^2 = a\). An error results if \(a\) is not a square.

Root(a, n): FldFinElt, RngIntElt -> FldFinElt

The \(n\)-th root of the non-zero element \(a\) from the field \(F\), i.e., an element \(y\) of \(F\) such that \(y^n = a\). An error results if no such root exists.

IsPower(a, n): FldFinElt, RngIntElt -> BoolElt, FldFinElt

Given a finite field element \(a\in F\), and an integer \(n>0\), this function returns either true and an element \(b\in F\) such that \(b^n=a\), or it returns false in the case that such an element does not exist.

AllRoots(a, n): FldFinElt, RngIntElt -> SeqEnum

Given a finite field element \(a\in F\), and an integer \(n>0\), return a sequence containing all of the \(n\)-th roots of \(a\) which lie in the same field \(F\).

Example: Roots and Traces of Elements (ex-ead4e7)

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Given the fields \(F\) and \(F49\) defined above, we can use the following functions:

> F7 := FiniteField(7);
> F49<w> := ext< F7 | 2 >;
> F<z> := ext< F49 | 2 >;
> Root(z^73, 7);
z^1039
> Trace(z^73);
1
> Trace(z^73, F49);
w^44
> Norm(z^73);
3
> Norm(z^73, F49);
w^37
> Norm(w^37);
3
> MinimalPolynomial(z^73);
x^2 + w^20*x + w^43
> MinimalPolynomial(z^73, F7);
x^4 + 4*x^2 + 4*x + 3

We now demonstrate the NormEquation function.

> Norm(z);
3
> NormEquation(F, F7!3);
true z
> Norm(z^30, F49);
w^30
> Parent(z) eq F;
true
> NormEquation(F, w^30);
true z^30

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