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See also Section Generic Element Functions.
+ 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 : FldFinElt, 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
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.
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.
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 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.
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 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.
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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