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: Functions (ex-ead4e7)#

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

Run in calculator

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

Run in calculator