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FINITE FIELDS

 
Acknowledgements
 
Introduction
      Representation of Finite Fields
      Conway Polynomials
      Ground Field and Relationships
 
Creation Functions
      Creation of Structures
      Creating Relations
      Special Options
      Homomorphisms
      Creation of Elements
      Special Elements
      Sequence Conversions
 
Structure Operations
      Related Structures
      Numerical Invariants
      Defining Polynomial
      Ring Predicates and Booleans
      Roots
 
Element Operations
      Arithmetic Operators
      Equality and Membership
      Parent and Category
      Predicates on Ring Elements
      Minimal and Characteristic Polynomial
      Norm, Trace and Frobenius
      Order and Roots
 
Polynomials for Finite Fields
 
Discrete Logarithms
 
Permutation Polynomials
 
Bibliography







DETAILS

 
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, 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 ]

 
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 (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)

 
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 (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 Mon Dec 17 14:40:36 EST 2012