Category(F) : FldFin -> Cat
Parent(F) : FldFin -> PowerStructure
Centre(F) : FldFin -> FldFin
PrimeRing(F) : FldFin -> FldFin
PrimeField(F) : FldFin -> FldFin
FieldOfFractions(F) : FldFin -> FldFin
Given F=GF(q), create the finite additive abelian group A of order
q=pr that is the direct sum of r copies of the cyclic group of order p,
together with the corresponding isomorphism from the group A to the field F.
UnitGroup(F) : FldFin -> GrpAb, Map
Given F=GF(q), create the multiplicative group of R as an abelian group.
This returns the (additive) cyclic group A of order q - 1, together
with a map from A to F - 0,
sending 1 to a primitive element of F.
Create the enumerated set consisting of the elements of finite field F.
Given a finite field F that is an extension of degree n of E,
define the natural isomorphism
between F and the n-dimensional vector
space En.
The function returns two values:
- (a)
- A vector space V isomorphic to En;
- (b)
- The isomorphism φ : F -> V.
The basis of V is chosen to correspond with the power
basis α0, α1, ..., αn - 1 of F,
where α is the generator returned by Generator(F, E),
so that V=E.1 x E.α x ... x E.αn - 1
and φ : αi -> ei + 1, (for i = 0, ..., n - 1),
where ei is the basis vector of V having all
components zero, except the i-th, which is one.
Given a finite field F that is an extension of degree n of E,
define the isomorphism between F and the n-dimensional vector space
En defined by the basis B for F over E.
The function returns two values:
- (a)
- A vector space V isomorphic to En;
- (b)
- The isomorphism φ : F -> V.
The basis of V is chosen to correspond with the basis B=β1, β2, ..., βn of F over E, as specified by the user,
so that
V=E.β1 x E.β2 x ... x E.βn.
φ : βi -> ei, (for i = 1, ..., n),
where ei is the basis vector of V having all
components zero, except the i-th, which is one.
Let F be a finite field that is an extension of degree n of E.
The function returns two values:
- (a)
- A matrix algebra A of degree n, such that A is isomorphic to F;
- (b)
- An isomorphism φ : F -> A.
The matrix algebra A will be the subalgebra of the full algebra
of n x n matrices over E generated by the companion
matrix C of the defining polynomial of F over E. The generator
Generator(F, E) of F over E is thus mapped to C.
Let F be a finite field. Let A be a matrix algebra over F,
and E be a subfield of F.
The function returns two values:
- (a)
- A matrix algebra N over E isomorphic to A, obtained from A
by expanding each component of an element of A into the block matrix associated with it;
- (b)
- An E-isomorphism φ : A -> N.
The matrix algebra N is A considered as an E-matrix algebra.
Given the field F of 7
4 elements defined as an extension of the
field F49 of 7
2 elements as above, we can construct two vector spaces,
one of dimension 2, and the other of dimension 4:
> F7 := FiniteField(7);
> F49<w> := ext< F7 | 2 >;
> F<z> := ext< F49 | 2 >;
> v2, i2 := VectorSpace(F, F49);
> v2;
Full Vector space of degree 2 over GF(7^2)
> i2(z^12);
( w w^28)
> v4, i4 := VectorSpace(F, PrimeField(F));
> v4;
Full Vector space of degree 4 over GF(7)
> i4(z^12);
(5 3 6 4)
Compute the Galois group (which is of course cyclic)
of K/k as a permutation group. The group is returned as well as the
roots of the defining polynomial of K/k in a compatible ordering.
Computes the (cyclic) group of k-automorphisms of K. The
group is returned as well as a sequence of all automorphisms and
a map sending an element of the abstract automorphism group to
an explicit automorphism.
Characteristic(F) : FldFin -> RngIntElt
# F : FldFin -> FldFinElt
The absolute degree of F, that is, the degree over its
prime subfield.
Given a finite field F that has been constructed as an
extension of a field E, return the degree of F over
E.
Given a finite field F that has been constructed as an
extension of a field E, return the polynomial with coefficients
in E that was used to define F as an extension of E.
This is the minimum polynomial of F.1.
Given a finite field F and a subfield E,
return the polynomial with coefficients
in E used to define F as an extension of E.
This is the same as the minimum polynomial of the generator
Generator(F, E) over E.
IsCommutative(F) : FldFin -> BoolElt
IsUnitary(F) : FldFin -> BoolElt
IsFinite(F) : FldFin -> BoolElt
IsOrdered(F) : FldFin -> BoolElt
IsField(F) : FldFin -> BoolElt
IsEuclideanDomain(F) : FldFin -> BoolElt
IsPID(F) : FldFin -> BoolElt
IsUFD(F) : FldFin -> BoolElt
IsDivisionRing(F) : FldFin -> BoolElt
IsEuclideanRing(F) : FldFin -> BoolElt
IsPrincipalIdealRing(F) : FldFin -> BoolElt
IsDomain(F) : FldFin -> BoolElt
F eq G : FldFin, Rng -> BoolElt
F ne G : FldFin, Rng -> BoolElt
Given a finite field F, this function returns true iff F is
defined over its prime field using a Conway polynomial.
Given a finite field F, this function returns true iff F is
a default field.
Given a polynomial f over a finite field F, this function finds all roots
of f in F, and returns a sorted sequence of tuples (pairs),
each consisting of a root of f in F and its multiplicity.
Given a univariate polynomial f over a finite field K, compute the
minimal splitting field S of f as an extension field of K, and return the
roots of f in S, together with S. Using this function will be
faster than computing the roots of f anew over the splitting field.
FactorisationOverSplittingField(f) : RngUPolElt[FldFin] -> [<RngUPolElt, RngIntElt>], FldFin
Given a univariate polynomial f over a finite field K, compute the
minimal splitting field S of f as an extension field of K, and return the
factorization (into linears) of f over S, together with S. Using
this function will be faster than factorizing f anew over the splitting
field.
Return a primitive n-th root of unity in the smallest possible extension
field of K.
We compute the roots of a certain degree-20 polynomial f in
its minimal splitting field.
> K := GF(2);
> P<x> := PolynomialRing(GF(2));
> f := x^20 + x^11 + 1;
> Factorization(f);
[
<x^3 + x^2 + 1, 1>,
<x^8 + x^7 + x^3 + x^2 + 1, 1>,
<x^9 + x^7 + x^6 + x^4 + 1, 1>
]
> time r, S<w> := RootsInSplittingField(f);
Time: 0.040
We note that the splitting field S has degree 72 and there are 20 roots
of f in S of course. We check that the evaluation of f at each root
is zero.
> S;
Finite field of size 2^72
> DefiningPolynomial(S);
x^72 + x^48 + x^47 + x^44 + x^38 + x^35 + x^32 + x^31 + x^30 +
x^29 + x^27 + x^25 + x^23 + x^22 + x^21 + x^18 + x^15 +
x^12 + x^8 + x^4 + 1
> #r;
20
> r[1];
<w^68 + w^67 + w^64 + w^62 + w^60 + w^59 + w^56 + w^50 + w^49 +
w^48 + w^47 + w^44 + w^43 + w^39 + w^37 + w^35 + w^33 + w^32
+ w^30 + w^29 + w^28 + w^25 + w^21 + w^19 + w^18 + w^16 +
w^15 + w^14 + w^12 + w^10 + w^6 + w, 1>
> [IsZero(Evaluate(f, t[1])): t in r];
[ true, true, true, true, true, true, true, true, true, true, true, true,
true, true, true, true, true, true, true, true ]
V2.28, 13 July 2023