Representation of Finite Fields
Ground Field and Relationships
Creation of Structures
FiniteField(q) : RngIntElt -> FldFin
FiniteField(p, n) : RngIntElt, RngIntElt -> FldFin
ext<F | n> : FldFin, RngIntElt -> FldFin, Map
ext<F | P> : FldFin, RngUPolElt[FldFin] -> FldFin, Map
ExtensionField<F, x | P> : FldFin, ... -> FldFin, Map
RandomExtension(F, n) : FldFin, RngIntElt -> FldFin
SplittingField(P) : RngUPolElt[FldFin] -> FldFin
SplittingField(S) : RngUPolElt[FldFin] -> FldFin
sub<F | d> : FldFin, RngIntElt -> FldFin, Map
sub<F | f> : FldFin, RngUPolElt[FldFin] -> FldFin, Map
GroundField(F) : FldFin -> FldFin
PrimeField(F) : FldFin -> FldFin
IsPrimeField(F) : Fld -> BoolElt
F meet G : FldFin, FldFin -> FldFin
CommonOverfield(K, L) : FldFin, FldFin -> FldFin
Example FldFin_Extensions (H21E1)
Creating Relations
Embed(E, F) : FldFin, FldFin ->
Embed(E, F, x) : FldFin, FldFin ->
Special Options
AssertAttribute(FldFin, "PowerPrinting", l) : Cat, MonStgElt, BoolElt ->
SetPowerPrinting(F, l) : FldFin, BoolElt ->
HasAttribute(FldFin, "PowerPrinting", l) : Cat, MonStgElt, BoolElt ->
HasAttribute(F, "PowerPrinting") : FldFin, MonStgElt -> BoolElt, BoolElt
AssignNames(~F, [f]) : FldFin, [ MonStgElt ]) ->
Name(F, 1) : FldFin, RngIntElt -> FldFinElt
Homomorphisms
hom< F -> G | x > : FldFin, Rng -> Map
Creation of Elements
F . 1 : FldFin -> FldFinElt
elt<F | a> : FldFin, RngElt -> FldFinElt
elt<F | a0, ..., an - 1> : FldFin, [FldFinElt] -> FldFinElt
Random(F) : FldFin -> FldFinElt
Special Elements
F . 1 : FldFin, RngIntElt -> FldFinElt
Generator(F, E) : FldFin, FldFin -> FldFinElt
PrimitiveElement(F) : FldFin -> FldFinElt
SetPrimitiveElement(F, x) : FldFin, FldFinElt ->
NormalElement(F) : FldFin -> FldFinElt
NormalElement(F, E) : FldFin, FldFin -> FldFinElt
Sequence Conversions
SequenceToElement(s, F) : [ FldFinElt ] -> FldFinElt
ElementToSequence(a) : FldFinElt -> [ FldFinElt ]
ElementToSequence(a, E) : FldFinElt, FldFin -> [ FldFinElt ]
Related Structures
AdditiveGroup(F) : FldFin -> GrpAb, Map
MultiplicativeGroup(F) : FldFin -> GrpAb, Map
Set(F) : FldFin -> SetEnum
VectorSpace(F, E) : FldFin, FldFin -> ModTupFld, Map
VectorSpace(F, E, B) : FldFin, FldFin, [ FldFinElt ] -> ModTupFld, Map
MatrixAlgebra(F, E) : FldFin, FldFin -> AlgMat, Map
MatrixAlgebra(A, E) : AlgMat, FldFin -> AlgMat, Map
Example FldFin_VectorSpace (H21E2)
GaloisGroup(K, k) : FldFin, FldFin -> GrpPerm, [FldFinElt]
AutomorphismGroup(K, k) : FldFin, FldFin -> GrpPerm, [Map], Map
Numerical Invariants
Degree(F) : FldFin -> RngIntElt
Degree(F, E) : FldFin, FldFin -> RngIntElt
Defining Polynomial
DefiningPolynomial(F) : FldFin -> RngUPolElt
DefiningPolynomial(F, E) : FldFin -> RngUPolElt
Ring Predicates and Booleans
IsConway(F) : FldFin -> BoolElt
IsDefault(F) : FldFin -> BoolElt
Roots
Roots(f) : RngUPolElt -> [ < FldFinElt, RngIntElt> ]
RootsInSplittingField(f) : RngUPolElt[FldFin] -> [<RngUPolElt, RngIntElt>], FldFin
FactorizationOverSplittingField(f) : RngUPolElt[FldFin] -> [<RngUPolElt, RngIntElt>], FldFin
RootOfUnity(n, K) : RngIntElt, FldFin -> FldFinElt
Example FldFin_Functions (H21E3)
Predicates on Ring Elements
IsPrimitive(a) : FldFinElt -> BoolElt
IsPrimitive(f) : RngUPolElt -> BoolElt
IsNormal(a) : FldFinElt -> BoolElt
IsNormal(a, E) : FldFinElt -> BoolElt
IsSquare(a) : FldFinElt -> BoolElt
Minimal and Characteristic Polynomial
MinimalPolynomial(a) : FldFinElt -> RngUPolElt
MinimalPolynomial(a, E) : FldFinElt, FldFin -> RngUPolElt
CharacteristicPolynomial(a) : FldFinElt -> RngUPolElt
CharacteristicPolynomial(a, E) : FldFinElt, FldFin -> RngUPolElt
Norm, Trace and Frobenius
Norm(a) : FldFinElt -> FldFinElt
Norm(a, E) : FldFinElt, FldFin -> FldFinElt
AbsoluteNorm(a) : FldFinElt -> FldFinElt
Trace(a) : FldFinElt -> FldFinElt
Trace(a, E) : FldFinElt, FldFin -> FldFinElt
AbsoluteTrace(a) : FldFinElt -> FldFinElt
Frobenius(a) : FldFinElt -> FldFinElt
Frobenius(a, r) : FldFinElt, RngIntElt -> FldFinElt
Frobenius(a, E) : FldFinElt, FldFin -> FldFinElt
Frobenius(a, E, r) : FldFinElt, FldFin, RngIntElt -> FldFinElt
NormEquation(K, y) : FldFin, FldFin -> BoolElt, FldFinElt
Hilbert90(a, q) : FldFinElt, RngIntElt -> FldFinElt
AdditiveHilbert90(a, q) : FldFinElt, RngIntElt -> FldFinElt
Order and Roots
Order(a) : FldFinElt -> RngIntElt
FactoredOrder(a) : FldFinElt -> RngIntElt
SquareRoot(a) : FldFinElt -> FldFinElt
Root(a, n) : FldFinElt, RngIntElt -> FldFinElt
IsPower(a, n) : FldFinElt, RngIntElt -> BoolElt, FldFinElt
AllRoots(a, n) : FldFinElt, RngIntElt -> SeqEnum
Example FldFin_Functions (H21E4)
Polynomials for Finite Fields
IrreduciblePolynomial(F, n) : FldFin, RngIntElt -> RngUPolElt
RandomIrreduciblePolynomial(F, n) : FldFin, RngIntElt -> RngUPolElt
IrreducibleLowTermGF2Polynomial(n) : RngIntElt -> RngUPolElt
IrreducibleSparseGF2Polynomial(n) : RngIntElt -> RngUPolElt
PrimitivePolynomial(F, m) : FldFin, RngIntElt -> RngUPolElt
AllIrreduciblePolynomials(F, m) : FldFin, RngIntElt -> { RngUPolElt }
ConwayPolynomial(p, n) : RngIntElt, RngIntElt -> RngUPolElt
ExistsConwayPolynomial(p, n) : RngIntElt, RngIntElt -> BoolElt, RngUPolElt
Discrete Logarithms
Log(x) : FldFinElt -> RngIntElt
Log(b, x) : FldFinElt, FldFinElt -> RngIntElt
ZechLog(K, n) : FldFin, RngIntElt -> RngIntElt
Sieve(K) : FldFin ->
SetVerbose("FFLog", v) : MonStgElt, RngIntElt ->
Example FldFin_Log (H21E5)
Permutation Polynomials
DicksonFirst(n, a) : RngIntElt, RngElt -> RngUPolElt
DicksonSecond(n, a) : RngIntElt, RngElt -> RngUPolElt
IsProbablyPermutationPolynomial(p) : RngUPolElt -> BoolElt
Example FldFin_Dickson (H21E6)
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013