Introduction

      Representation of Finite Fields

      Conway Polynomials

      Ground Field and Relationships

 
Creation Functions

      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, FldFinElt -> 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 (H22E1)

      Creating Relations
            Embed(E, F) : FldFin, FldFin ->
            Embed(E, F, x) : FldFin, FldFin ->
            IsIsomorphic(E, F) : FldFin, FldFin -> BoolElt, Map[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, ..., a_n - 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 ]

 
Structure Operations

      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 (H22E2)
            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 (H22E3)

 
Element Operations

      Arithmetic Operators

      Equality and Membership

      Parent and Category

      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 (H22E4)

 
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 (H22E5)

 
Permutation Polynomials
      DicksonFirst(n, a) : RngIntElt, RngElt -> RngUPolElt
      DicksonSecond(n, a) : RngIntElt, RngElt -> RngUPolElt
      IsProbablyPermutationPolynomial(p) : RngUPolElt -> BoolElt
      Example FldFin_Dickson (H22E6)

 
Bibliography

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