Subsections

• Arithmetic Operators
• Equality and Membership
• Parent and Category
• Predicates on Ring Elements
• Minimal and Characteristic Polynomial
• Norm and Trace
• Log, Order and Roots

+ 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

a eq b : FldFinElt, FldFinElt -> BoolElt

a ne b : FldFinntRest, FldFinElt -> BoolElt

a in F : FldFinElt, Rng -> BoolElt

a notin F : FldFinElt, Rng -> BoolElt

Parent(a) : FldFinElt -> FldFin

Category(a) : FldFinElt -> Cat

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

True if and only if the element a of F is a primitive element for F.

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).

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, ..., a^(q^(n - 1)) form a basis for F over the ground field G=GF(q).

IsNormal(a, E) : FldFinElt -> BoolElt

True if and only if the element a of F=GF(q^n) generates a normal basis for F over its subfield E=GF(q), that is, if and only if a, a^q, ..., a^(q^(n - 1)) form a basis for F over E.

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.

The minimal polynomial of the element a of the field F, relative to the ground field of F. This is a polynomial with coefficients in the ground field, having a as a root.

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

The minimal polynomial of the element a of the field F, relative to the subfield E of F. This is a polynomial with coefficients in E, having a as a root.

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.)

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.

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.

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.

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.

The discrete logarithm of a non-zero element a from the field F, i.e., an integer k such that a = w^k, where w is the primitive element of F (as returned by PrimitiveElement).

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

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

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.

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.

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.

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

> F7 := FiniteField(7);
> F49<w> := ext< F7 | 2 >;
> F<z> := ext< F49 | 2 >;
> Root(z^73, 7);
z^1039
> Trace(z^73);
0
> 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