See also Section Generic Element Functions.
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).
Given a univariate polynomial f∈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).
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, aq, ..., a^(qn - 1) form a basis for F over the ground field G=GF(q).
Returns true if and only if the element a of the finite field F with qn elements generates a normal basis for F over its subfield E, that is, if and only if a, aq, ..., a^(qn - 1) form a basis for F over E for q=#E.
Given a finite field element a∈F, this function returns either true and an element b∈F such that b2=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 the unique minimal-degree monic polynomial with coefficients in the ground field, having a as a root.
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.
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.)
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.
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.
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.
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.
The trace of the element a, that is, the trace to the prime subfield of the parent field F of a.
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.
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.
The Frobenius image of x w.r.t. E; i.e., x#E.
The Frobenius image of x w.r.t. E; i.e., x(#E)r.
Given a finite field K and an element y of a subfield S of K, return whether an element x∈K exists such that Norm(x, S) = y, and, if so, such an element x (in K).
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 xqx - 1=a. Note that the solution may be in a finite-degree extension of k.
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 xq - x=a. Note that the solution may be in a finite-degree extension of k.
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.
The square root of the non-zero element a from the field F, i.e., an element y of F such that y2 = 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 yn = a. An error results if no such root exists.
Given a finite field element a∈F, and an integer n>0, this function returns either true and an element b∈F such that bn=a, or it returns false in the case that such an element does not exist.
Given a finite field element a∈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.
> 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 + 3We 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