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ALGEBRAIC FUNCTION FIELDS

 
Acknowledgements
 
Introduction
      Representations of Fields
 
Creation of Algebraic Function Fields and their Orders
      Creation of Algebraic Function Fields
      Creation of Orders of Algebraic Function Fields
      Orders and Ideals
 
Related Structures
      Parent and Category
      Other Related Structures
 
General Structure Invariants
 
Galois Groups
 
Subfields
 
Automorphism Group
      Automorphisms over the Base Field
      General Automorphisms
      Field Morphisms
 
Global Function Fields
      Functions relative to the Exact Constant Field
      Functions Relative to the Constant Field
      Functions related to Class Group
 
Structure Predicates
 
Homomorphisms
 
Elements
      Creation of Elements
      Parent and Category
      Sequence Conversions
      Arithmetic Operators
      Equality and Membership
      Predicates on Elements
      Functions related to Norm and Trace
      Functions related to Orders and Integrality
      Functions related to Places and Divisors
      Other Operations on Elements
 
Ideals
      Creation of Ideals
      Parent and Category
      Arithmetic Operators
      Roots of Ideals
      Equality and Membership
      Predicates on Ideals
            Predicates on Prime Ideals
      Further Ideal Operations
            Functions on Prime Ideals
 
Places
      Creation of Structures
      Creation of Elements
            General Function Field Places
            Global Function Field Places
      Related Structures
            Parent and Category
      Structure Invariants
            General function fields
            Global Function Fields
      Structure Predicates
      Element Operations
            Parent and Category
            Arithmetic Operators
            Equality and Membership
            Predicates on Elements
            Other Element Operations
      Completion at Places
 
Divisors
      Creation of Structures
      Creation of Elements
      Related Structures
            Parent and Category
      Structure Invariants
      Structure Predicates
      Element Operations
            Arithmetic Operators
            Equality, Comparison and Membership
            Predicates on Elements
            Other Element Operations
      Functions related to Divisor Class Groups of Global Function Fields
 
Differentials
      Creation of Structures
      Creation of Elements
      Related Structures
      Subspaces
      Structure Predicates
      Operations on Elements
            Arithmetic Operators
            Equality and Membership
            Predicates on Elements
            Functions on Elements
            Other
 
Weil Descent
 
Function Field Database
      Creation
      Access
 
Bibliography







DETAILS

 
Introduction

      Representations of Fields

 
Creation of Algebraic Function Fields and their Orders

      Creation of Algebraic Function Fields
            ext< K | f > : FldFunRat, RngUPolElt -> FldFun
            FunctionField(f : parameters) : RngMPolElt -> FldFun
            FunctionField(S) : [RngUPolElt] -> FldFun
            FunctionField(S) : [RngMPolElt] -> FldFun
            HermitianFunctionField(p, d) : RngIntElt, RngIntElt -> FldFun
            sub<F | S> : FldFun, [] -> FldFun
            AssignNames(~F, s) : FldFun, [ MonStgElt ] ->
            FunctionField(R) : Rng -> FldFunG
            Example FldFunG_Creation (H42E1)
            Example FldFunG_creation-rel (H42E2)
            Example FldFunG_creation-non-simple (H42E3)
            Example FldFunG_creation_herm (H42E4)

      Creation of Orders of Algebraic Function Fields
            EquationOrderFinite(F) : FldFun -> RngFunOrd
            MaximalOrderFinite(F) : FldFun -> RngFunOrd
            EquationOrderInfinite(F) : FldFun -> RngFunOrd
            MaximalOrderInfinite(F) : FldFun -> RngFunOrd
            IntegralClosure(R, F) : Rng, FldFun -> RngFunOrd
            EquationOrder(O) : RngFunOrd -> RngFunOrd
            MaximalOrder(O) : RngFunOrd -> RngFunOrd
            SetOrderMaximal(O, b) : RngFunOrd, BoolElt ->
            ext<O | f> : RngFunOrd, RngUPolElt -> RngFunOrd
            Example FldFunG_orders (H42E5)
            Example FldFunG_int_cl (H42E6)
            Order(O, T, d) : RngFunOrd, AlgMatElt, RngElt -> RngFunOrd
            Order(O, M) : RngFunOrd, ModDed -> RngFunOrd
            Order(O, S) : RngFunOrd, [FldFunElt] -> RngFunOrd
            Simplify(O) : RngFunOrd -> RngFunOrd
            O1 + O2 : RngFunOrd, RngFunOrd -> RngFunOrd
            O1 meet O2 : RngFunOrd, RngFunOrd -> RngFunOrd
            AsExtensionOf(O1, O2) : RngFunOrd, RngFunOrd -> RngFunOrd
            Example FldFunG_order-create-more (H42E7)

      Orders and Ideals
            MultiplicatorRing(I) : RngFunOrdIdl -> RngFunOrd
            pMaximalOrder(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrd
            pRadical(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl

 
Related Structures

      Parent and Category

      Other Related Structures
            PrimeRing(F) : FldFun -> Rng
            ConstantField(F) : FldFunG -> Rng
            ExactConstantField(F) : FldFunG -> Rng, Map
            BaseRing(F) : FldFun -> Rng
            ISABaseField(F,G) : Fld, Fld -> BoolElt
            BaseRing(O) : RngFunOrd -> Rng
            BaseRing(FF) : FldFunOrd -> Rng
            SubOrder(O) : RngFunOrd -> RngFunOrd
            FunctionField(O) : RngFunOrd -> FldFun
            FieldOfFractions(O) : RngFunOrd -> FldFunOrd
            Order(FF) : FldFunOrd -> RngFunOrd
            RationalExtensionRepresentation(F) : FldFunG -> FldFun
            AbsoluteOrder(O) : RngFunOrd -> RngFunOrd
            AbsoluteFunctionField(F) : FldFunG -> FldFunG
            UnderlyingRing(F) : FldFunG -> FldFunG
            Embed(F, L, a) : FldFun, FldFun, FldFunElt ->
            Places(F) : FldFunG -> PlcFun
            DivisorGroup(F) : FldFun -> DivFun
            DifferentialSpace(F) : FldFun -> DiffFun
            Example FldFunG_related-structures (H42E8)
            Example FldFunG_related-structures-rat-ext (H42E9)
            WeilRestriction(E, n) : FldFun, RngIntElt -> FldFun, UserProgram
            ConstantFieldExtension(F, E) : FldFun, Rng -> FldFun, Map
            Example FldFunG_cfe (H42E10)
            Reduce(O) : RngFunOrd -> RngFunOrd

 
General Structure Invariants
      Characteristic(F) : FldFun -> RngIntElt
      IsPerfect(F) : Fld -> BoolElt
      Degree(F) : FldFunG -> RngIntElt
      AbsoluteDegree(F) : FldFunG -> RngIntElt
      DefiningPolynomial(F) : FldFun -> RngUPolElt
      DefiningPolynomials(F) : FldFun -> [RngUPolElt]
      Basis(F) : FldFunG -> SeqEnum[FldFunElt]
      TransformationMatrix(O1, O2) : RngFunOrd, RngFunOrd -> AlgMatElt, RngElt
      CoefficientIdeals(O) : RngFunOrd -> [RngFunOrdIdl]
      BasisMatrix(O) : RngFunOrd -> AlgMatElt
      PrimitiveElement(O) : RngFunOrd -> RngFunOrdElt
      Discriminant(O) : RngFunOrd -> .
      AbsoluteDiscriminant(O) : RngFunOrd -> .
      DimensionOfExactConstantField(F) : FldFunG -> RngIntElt
      Genus(F) : FldFunG -> RngIntElt
      Example FldFunG_invar (H42E11)
      Example FldFunG_invar-non-simple (H42E12)
      GapNumbers(F) : FldFunG -> SeqEnum[RngIntElt]
      GapNumbers(F, P) : FldFunG, PlcFunElt -> SeqEnum[RngIntElt]
      SeparatingElement(F) : FldFunG -> FldFunGElt
      RamificationDivisor(F) : FldFunG -> DivFunElt
      WeierstrassPlaces(F) : FldFunG -> [PlcFunElt]
      WronskianOrders(F) : FldFunG -> [RngIntElt]
      Different(O) : RngFunOrd -> RngFunOrdIdl
      Index(O, S) : RngFunOrd, RngFunOrd -> Any

 
Galois Groups
      GaloisGroup(f) : RngUPolElt -> GrpPerm, [ RngElt ], GaloisData
      GaloisGroup(F) : FldFun -> GrpPerm, [RngElt], GaloisData
      Example FldFunG_GaloisGroups (H42E13)
      Example FldFunG_GaloisGroups2 (H42E14)

 
Subfields
      Subfields(F) : FldFun -> SeqEnum[FldFun]
      Example FldFunG_Subfields (H42E15)

 
Automorphism Group

      Automorphisms over the Base Field
            Automorphisms(K, k) : FldFun, FldFunG -> [Map]
            AutomorphismGroup(K, k) : FldFun, FldFunG -> GrpFP, Map
            Example FldFunG_Automorphisms (H42E16)
            IsSubfield(K, L) : FldFun, FldFun -> BoolElt, Map
            IsIsomorphicOverQt(K, L) : FldFun, FldFun -> BoolElt, Map
            Example FldFunG_IsSubfield (H42E17)

      General Automorphisms
            Isomorphisms(K, E) : FldFunG, FldFunG -> [Map]
            IsIsomorphic(K, E) : FldFunG, FldFunG -> BoolElt, Map
            Automorphisms(K) : FldFunG -> [Map]
            Isomorphisms(K,E,p1,p2) : FldFunG, FldFunG, PlcFunElt, PlcFunElt -> [Map]
            AutomorphismGroup(K) : FldFunG -> GrpFP, Map
            AutomorphismGroup(K,f) : FldFunG, Map -> Grp, Map, SeqEnum

      Field Morphisms
            IsMorphism(f) : Map -> Bool
            FieldMorphism(f) : Map -> Map
            IdentityFieldMorphism(F) : Fld -> Map
            IsIdentity(f) : Map -> BoolElt
            Equality(f, g) : Map, Map -> Bool
            HasInverse(f) : Map -> MonStgElt, Map
            Composition(f, g) : Map, Map -> Map
            Example FldFunG_Isomorphisms (H42E18)

 
Global Function Fields

      Functions relative to the Exact Constant Field
            NumberOfPlacesOfDegreeOverExactConstantField(F, m) : FldFun, RngIntElt -> RngIntElt
            NumberOfPlacesOfDegreeOneOverExactConstantField(F) : FldFunG -> RngIntElt
            NumberOfPlacesOfDegreeOneOverExactConstantField(F, m) : FldFunG, RngIntElt -> RngIntElt
            SerreBound(F) : FldFunG -> RngIntElt
            IharaBound(F) : FldFunG -> RngIntElt
            NumberOfPlacesOfDegreeOneECFBound(F) : FldFunG -> RngIntElt
            LPolynomial(F) : FldFunG -> RngUPolElt
            LPolynomial(F, m) : FldFunG, RngIntElt -> RngUPolElt
            ZetaFunction(F) : FldFunG -> FldFunRatUElt
            ZetaFunction(F, m) : FldFunG, RngIntElt -> FldFunRatUElt

      Functions Relative to the Constant Field
            Places(F, m) : FldFunG, RngIntElt -> SeqEnum[PlcFunElt]
            HasPlace(F, m) : FldFunG, RngIntElt -> BoolElt, PlcFunElt
            HasRandomPlace(F, m) : FldFunG, RngIntElt -> BoolElt, PlcFunElt
            RandomPlace(F, m) : FldFunG, RngIntElt -> PlcFunElt
            Example FldFunG_global-function-fields (H42E19)
            Example FldFunG_global1 (H42E20)

      Functions related to Class Group
            UnitRank(O) : RngFunOrd -> RngIntElt
            UnitGroup(O) : RngFunOrd -> GrpAb, Map
            Regulator(O) : RngFunOrd -> RngIntElt
            PrincipalIdealMap(O) : RngFunOrd -> Map
            Example FldFunG_global-class-ex (H42E21)
            ClassGroup(F : parameters) : FldFunG -> GrpAb, Map, Map
            ClassGroup(O) : RngFunOrd -> GrpAb, Map, Map
            ClassGroupExactSequence(O) : RngFunOrd -> Map, Map, Map
            ClassGroupAbelianInvariants(F : parameters) : FldFunG -> SeqEnum
            ClassGroupAbelianInvariants(O) : RngFunOrd -> SeqEnum
            ClassNumber(F) : FldFunG -> RngIntElt
            ClassNumber(O) : RngFunOrd -> RngIntElt
            Example FldFunG_class-group (H42E22)
            GlobalUnitGroup(F) : FldFunG -> GrpAb, Map
            ClassGroupPRank(F) : FldFunG -> RngIntElt
            HasseWittInvariant(F) : FldFunG -> RngIntElt
            IndependentUnits(O) : RngFunOrd -> SeqEnum[RngFunOrdElt]
            FundamentalUnits(O) : RngFunOrd -> SeqEnum[RngFunOrdElt]
            Example FldFunG_orders (H42E23)

 
Structure Predicates
      O1 subset O2 : RngFunOrd, RngFunOrd -> BoolElt
      IsGlobal(F) : FldFunG -> BoolElt
      IsRationalFunctionField(F) : FldFunG -> BoolElt
      IsFiniteOrder(O) : RngFunOrd -> BoolElt
      IsEquationOrder(O) : RngFunOrd -> BoolElt
      IsAbsoluteOrder(O) : RngFunOrd -> BoolElt
      IsMaximal(O) : RngFunOrd -> BoolElt
      IsTamelyRamified(O) : RngFunOrd -> BoolElt
      IsTotallyRamified(O) : RngFunOrd -> BoolElt
      IsUnramified(O) : RngFunOrd -> BoolElt
      IsWildlyRamified(O) : RngFunOrd -> BoolElt
      IsInKummerRepresentation(K) : FldFun -> BoolElt, FldFunElt
      IsInArtinSchreierRepresentation(K) : FldFun -> BoolElt, FldFunElt

 
Homomorphisms
      hom<F -> R | g> : FldFun, Rng, RngElt -> Map
      hom< O -> R | g > : RngFunOrd, Rng, RngElt -> Map
      IsRingHomomorphism(m) : Map -> BoolElt
      Example FldFunG_hom (H42E24)
      hom< O -> R | b1, ..., bn > : RngFunOrd, Rng, RngElt, ..., RngElt -> Map

 
Elements

      Creation of Elements
            F . 1 : FldFun -> FldFunElt
            Name(F, i) : FldFun, RngIntElt -> FldFunElt
            O . i : RngFunOrd, RngIntElt -> FldFunOrdElt
            F ! a : FldFun, . -> FldFunElt
            O ! a : RngFunOrd, . -> RngFunOrdElt
            FF ! a : FldFunOrd, Any -> FldFunOrdElt
            elt< F | a0, a1, ..., an - 1> : FldFun, RngElt , ..., RngElt -> FldFunElt
            elt< O | a1, a2, ..., an> : RngFunOrd, RngElt , ..., RngElt -> RngFunOrdElt
            Random(F, m) : FldFunG, RngIntElt -> FldFunElt

      Parent and Category

      Sequence Conversions
            ElementToSequence(a) : FldFunElt -> SeqEnum[FldElt]
            Eltseq(a, R) : FldFunElt, FldFunG -> [FldFunGElt]
            Flat(a) : FldFunElt -> [FldFunGElt]
            F ! [ a0, a1, ..., an - 1 ] : FldFun, SeqEnum -> FldFunElt
            O ! [ a1, a2, ..., an ] : RngFunOrd, SeqEnum -> RngFunOrdElt
            Example FldFunG_Elements (H42E25)

      Arithmetic Operators
            Modexp(a, k, m) : RngFunOrdElt, RngIntElt, RngUPolElt -> RngFunOrdElt
            a mod I : RngFunOrdElt, RngFunOrdIdl -> RngFunOrdElt
            Modinv(a, m) : RngFunOrdElt, RngFunOrdIdl -> RngFunOrdElt

      Equality and Membership

      Predicates on Elements
            IsDivisibleBy(a, b) : FldFunElt, FldFunElt -> BoolElt, FldFunElt
            IsSeparating(a) : FldFunGElt -> BoolElt
            IsConstant(a) : FldFunGElt -> BoolElt, RngElt
            IsGlobalUnit(a) : FldFunElt -> BoolElt
            IsGlobalUnitWithPreimage(a) : FldFunElt -> BoolElt, GrpAbElt
            IsUnitWithPreimage(a) : RngFunOrdElt -> BoolElt, GrpAbElt

      Functions related to Norm and Trace
            RepresentationMatrix(a) : FldFunGElt -> AlgMatElt
            Trace(a, R) : FldFunElt, Rng -> RngElt
            Norm(a, R) : FldFunElt, Rng -> RngElt
            CharacteristicPolynomial(a, R) : FldFunElt, Rng -> RngUPolElt
            MinimalPolynomial(a, R) : FldFunElt, Rng -> RngUPolElt
            AbsoluteMinimalPolynomial(a) : FldFunElt -> RngUPolElt
            RepresentationMatrix(a, R) : FldFunGElt, Rng -> AlgMatElt
            Example FldFunG_elements-norm-trace (H42E26)

      Functions related to Orders and Integrality
            IntegralSplit(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt, RngElt
            Numerator(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt
            Numerator(a) : FldFunOrdElt -> RngFunOrdElt
            Numerator(a, O) : FldFunOrdElt, RngFunOrd -> RngElt
            Denominator(a, O) : FldFunElt, RngFunOrd -> RngElt
            Denominator(a) : FldFunOrdElt -> RngElt
            Denominator(a, O) : FldFunOrdElt, RngFunOrd -> RngElt
            Min(a, O) : FldFunElt, RngFunOrd -> RngElt, RngElt

      Functions related to Places and Divisors
            Evaluate(a, P) : FldFunElt, PlcFunElt -> RngElt
            Lift(a, P) : RngElt, PlcFunElt -> FldFunElt
            Valuation(a, P) : FldFunElt, PlcFunElt -> RngIntElt
            Expand(a, P) : FldFunGElt, PlcFunElt -> RngSerElt, FldFunGElt
            Divisor(a) : FldFunGElt -> DivFunElt
            Zeros(a) : FldFunGElt -> [PlcFunElt]
            Zeros(F, a) : FldFunG, FldFunGElt -> [PlcFunElt]
            Poles(a) : FldFunGElt -> SeqEnum[PlcFunElt]
            Poles(F, a) : FldFun, FldFunGElt -> [PlcFunElt]
            Degree(a) : FldFunElt -> RngIntElt
            CommonZeros(L) : [FldFunGElt] -> [PlcFunElt]
            CommonZeros(F, L) : FldFunG, SeqEnum[ FldFunGElt ] -> SeqEnum[ PlcFunElt ]
            Example FldFunG_elements (H42E27)
            Module(L, R) : SeqEnum[ FldFunGElt ], Rng -> Mod, Map, SeqEnum[ ModElt ]
            Relations(L, R) : SeqEnum[ FldFunElt ], Rng -> ModTupRng
            Roots(f, D) : RngUPolElt, DivFunElt -> SeqEnum[ FldFunElt ]
            Example FldFunG_module (H42E28)

      Other Operations on Elements
            ProductRepresentation(a) : FldFunGElt -> [FldFunGElt], [RngIntElt]
            ProductRepresentation(Q, S) : [FldFunGElt], [RngIntElt] -> FldFunGElt
            RationalFunction(a) : FldFunGElt -> RngElt
            Differentiation(x, a) : FldFunGElt, FldFunGElt -> FldFunGElt
            Differentiation(x, n, a) : FldFunGElt, RngIntElt, FldFunGElt -> FldFunGElt
            DifferentiationSequence(x, n, a) : FldFunGElt, RngIntElt, FldFunGElt -> SeqEnum
            PrimePowerRepresentation(x, k, a) : FldFunGElt, RngIntElt, FldFunGElt -> SeqEnum
            Different(a) : RngFunOrdElt -> RngFunOrdElt
            RationalReconstruction(e, f) : FldFunElt, RngUPolElt -> BoolElt, FldFunElt
            CoefficientHeight(a) : RngFunOrdElt -> RngIntElt
            CoefficientLength(a) : RngFunOrdElt -> RngIntElt
            Example FldFunG_elements-other_ops (H42E29)

 
Ideals

      Creation of Ideals
            ideal< O | a1, a2, ... , am > : RngFunOrd, RngElt, ..., RngElt -> RngFunOrdIdl
            ideal< O | T, d > : RngFunOrd, AlgMatElt, RngElt -> RngFunOrdIdl
            ideal< O | T, S > : RngFunOrd, AlgMatElt, [RngFunOrdIdl] -> RngFunOrdIdl
            x * O : RngElt, RngFunOrd -> RngFunOrdIdl
            Ideal(P) : PlcFunElt -> RngFunOrdIdl
            Ideals(D) : DivFunElt -> RngFunOrdIdl, RngFunOrdIdl
            O !! I : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl

      Parent and Category

      Arithmetic Operators
            c / I : RngElt, RngFunOrdIdl -> RngFunOrdIdl
            IdealQuotient(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
            ChineseRemainderTheorem(I1, I2, e1, e2) : RngFunOrdIdl, RngFunOrdIdl, RngFunOrdElt, RngFunOrdElt -> RngFunOrdElt

      Roots of Ideals
            IsPower(I, n) : RngFunOrdIdl, RngIntElt -> BoolElt, RngFunOrdIdl
            Root(I, n) : RngFunOrdIdl, RngIntElt -> RngFunOrdIdl
            IsSquare(I) : RngFunOrdIdl -> BoolElt, RngFunOrdIdl
            SquareRoot(I) : RngFunOrdIdl -> RngFunOrdIdl
            Example FldFunG_ideal-is-square (H42E30)

      Equality and Membership

      Predicates on Ideals
            IsZero(I) : RngFunOrdIdl -> BoolElt
            IsOne(I) : RngFunOrdIdl -> BoolElt
            IsIntegral(I) : RngFunOrdIdl -> BoolElt
            IsPrime(I) : RngFunOrdIdl -> BoolElt
            IsPrincipal(I) : RngFunOrdIdl -> BoolElt, FldFunElt

            Predicates on Prime Ideals
                  IsInert(P) : RngFunOrdIdl -> BoolElt
                  IsInert(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
                  IsRamified(P) : RngFunOrdIdl -> BoolElt
                  IsRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
                  IsSplit(P) : RngFunOrdIdl -> BoolElt
                  IsSplit(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
                  IsTamelyRamified(P) : RngFunOrdIdl -> BoolElt
                  IsTamelyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
                  IsTotallyRamified(P) : RngFunOrdIdl -> BoolElt
                  IsTotallyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
                  IsTotallySplit(P) : RngFunOrdIdl -> BoolElt
                  IsTotallySplit(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
                  IsUnramified(P) : RngFunOrdIdl -> BoolElt
                  IsUnramified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
                  IsWildlyRamified(P) : RngFunOrdIdl -> BoolElt
                  IsWildlyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt

      Further Ideal Operations
            I meet J : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
            Gcd(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
            Lcm(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
            Factorization(I) : RngFunOrdIdl -> [ <RngFunOrdIdl, RngIntElt> ]
            Decomposition(O, p) : RngFunOrd, RngElt -> [ RngFunOrdIdl ]
            Decomposition(O) : RngFunOrd -> [ RngFunOrdIdl ]
            DecompositionType(O, p) : RngFunOrd, RngElt -> [ <RngIntElt, RngIntElt> ]
            DecompositionType(O) : RngFunOrd -> [ <RngIntElt, RngIntElt> ]
            MultiplicatorRing(I) : RngFunOrdIdl -> RngFunOrd
            pMaximalOrder(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrd
            pRadical(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl
            Valuation(a, P) : RngElt, RngFunOrdIdl -> RngIntElt
            Order(I) : RngFunOrdIdl -> RngFunOrd
            Denominator(I) : RngFunOrdIdl -> RngElt
            Minimum(I) : RngFunOrdIdl -> Any
            I meet R : RngFunOrdIdl, Rng -> Any
            IntegralSplit(I) : RngFunOrdIdl -> RngFunOrdIdl, RngElt
            Norm(I) : RngFunOrdIdl -> Any
            TwoElement(I) : RngFunOrdIdl -> RngElt, RngElt
            Generators(I) : RngFunOrdIdl -> [ RngFunOrdElt ]
            Basis(I) : RngFunOrdIdl -> [FldFunElt]
            BasisMatrix(I) : RngFunOrdIdl -> AlgMatElt
            TransformationMatrix(I) : RngFunOrdIdl -> AlgMatElt, RngElt
            CoefficientIdeals(I) : RngOrdFracIdl -> [RngOrdFracIdl]
            Different(I) : RngFunOrdIdl -> RngFunOrdIdl
            Codifferent(I) : RngFunOrdIdl -> RngFunOrdIdl
            Divisor(I) : RngFunOrdIdl -> DivFunElt
            Divisor(I, J) : RngFunOrdIdl, RngFunOrdIdl -> DivFunElt
            Example FldFunG_ideals (H42E31)

            Functions on Prime Ideals
                  RamificationIndex(I) : RngFunOrdIdl -> RngIntElt
                  Degree(I) : RngFunOrdIdl -> RngIntElt
                  ResidueClassField(I) : RngFunOrdIdl -> Rng, Map
                  Place(I) : RngFunOrdIdl -> PlcFunElt
                  SafeUniformizer(P) : RngFunOrdIdl -> RngFunOrdElt
                  Example FldFunG_order-ideals (H42E32)

 
Places

      Creation of Structures
            Places(F) : FldFun -> PlcFun

      Creation of Elements

            General Function Field Places
                  Decomposition(F, P) : FldFunG, PlcFunElt -> [ PlcFunElt ]
                  DecompositionType(F, P) : FldFun, PlcFunElt -> [ <RngIntElt, RngIntElt> ]
                  Zeros(a) : FldFunElt -> [ PlcFunElt ]
                  Poles(a) : FldFunElt -> [ PlcFunElt ]
                  S ! I : PlcFun, RngFunOrdIdl -> PlcFunElt
                  Support(D) : DivFunElt -> [ PlcFunElt ], [ RngIntElt ]
                  AssignNames(~P, s) : PlcFunElt, [ MonStgElt ] ->
                  InfinitePlaces(F) : FldFun -> [PlcFunElt]

            Global Function Field Places
                  HasPlace(F, m) : FldFun, RngIntElt -> PlcFunElt
                  HasRandomPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
                  RandomPlace(F, m) : FldFun, RngIntElt -> PlcFunElt
                  Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
                  Example FldFunG_place-creation (H42E33)

      Related Structures

            Parent and Category
                  FunctionField(S) : PlcFun -> FldFun
                  DivisorGroup(F) : FldFunG -> DivFun

      Structure Invariants

            General function fields
                  WeierstrassPlaces(F) : FldFunG -> [PlcFunElt]

            Global Function Fields
                  NumberOfPlacesOfDegreeOneOverExactConstantField(F, m) : FldFun, RngIntElt -> RngIntElt
                  NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F, m) : FldFun, RngIntElt -> RngIntElt
                  NumberOfPlacesOfDegreeOverExactConstantField(F, m) : FldFunG, RngIntElt -> RngIntElt

      Structure Predicates

      Element Operations

            Parent and Category

            Arithmetic Operators
                  Quotrem(P, k) : PlcFunElt, RngIntElt -> DivFunElt, DivFunElt

            Equality and Membership

            Predicates on Elements
                  IsFinite(P) : PlcFunElt -> BoolElt
                  IsWeierstrassPlace(P) : PlcFunElt -> BoolElt

            Other Element Operations
                  FunctionField(P) : PlcFunElt -> FldFun
                  Degree(P) : PlcFunElt -> RngIntElt
                  RamificationIndex(P) : PlcFunElt -> RngIntElt
                  InertiaDegree(P) : PlcFunElt -> RngIntElt
                  Minimum(P) : PlcFunElt -> RngElt
                  ResidueClassField(P) : PlcFunElt -> Rng, Map
                  Evaluate(a, P) : RngElt, PlcFunElt -> RngElt
                  Lift(a, P) : RngElt, PlcFunElt -> FldFunElt
                  TwoGenerators(P) : PlcFunElt -> FldFunGElt, FldFunGElt
                  LocalUniformizer(P) : PlcFunElt -> FldFunGElt
                  SafeUniformizer(P) :
                  Ideal(P) : PlcFunElt -> RngFunOrdIdl
                  Norm(P) : PlcFunElt -> DivFunElt
                  Example FldFunG_places (H42E34)

      Completion at Places
            Completion(F, p) : FldFun, PlcFunElt -> RngSerLaur, Map

 
Divisors

      Creation of Structures
            DivisorGroup(F) : FldFun -> DivFun

      Creation of Elements
            Divisor(P) : PlcFunElt -> DivFunElt
            Div ! a : DivFun, RngElt -> DivFunElt
            Div ! I : DivFun, RngFunOrdIdl -> DivFunElt
            Divisor(I, J) : RngFunOrdIdl, RngFunOrdIdl -> DivFunElt
            Identity(G) : DivFun -> DivFunElt
            CanonicalDivisor(F) : FldFunG -> DivFunElt
            DifferentDivisor(F) : FldFunG -> DivFunElt
            AssignNames(~D, s) : DivFunElt, [ MonStgElt ] ->

      Related Structures

            Parent and Category
                  FunctionField(G) : DivFun -> FldFun
                  Places(F) : FldFun -> PlcFun

      Structure Invariants
            NumberOfSmoothDivisors(n, m, P) : RngIntElt, RngIntElt, SeqEnum[RngElt] -> RngElt
            DivisorOfDegreeOne(F) : FldFunG -> DivFunElt

      Structure Predicates

      Element Operations

            Arithmetic Operators
                  Quotrem(D, k) : DivFunElt, RngIntElt -> DivFunElt, DivFunElt
                  GCD(D1, D2) : DivFunElt, DivFunElt -> DivFunElt
                  LCM(D1, D2) : DivFunElt, DivFunElt -> DivFunElt

            Equality, Comparison and Membership

            Predicates on Elements
                  IsCanonical(D) : DivFunElt -> BoolElt, DiffFunElt
                  Example FldFunG_divisors-simple_rel (H42E35)

            Other Element Operations
                  FunctionField(D) : DivFunElt -> FldFun
                  Degree(D) : DivFunElt -> RngIntElt
                  Support(D) : DivFunElt -> [ PlcFunElt ]
                  Numerator(D) : DivFunElt -> DivFunElt
                  Denominator(D) : DivFunElt -> DivFunElt
                  Ideals(D) : DivFunElt -> RngFunOrdIdl, RngFunOrdIdl
                  Norm(D) : DivFunElt -> DivFunElt
                  FiniteSplit(D) : DivFunElt -> DivFunElt, DivFunElt
                  Dimension(D) : DivFunElt -> RngIntElt
                  IndexOfSpeciality(D) : DivFunElt -> RngIntElt
                  ShortBasis(D : parameters) : DivFunElt -> [RngElt], [RngIntElt]
                  Basis(D : parameters) : DivFunElt -> [ FldFunElt ]
                  RiemannRochSpace(D) : DivFunElt -> ModFld, Map
                  Valuation(D, P) : DivFunElt, PlcFunElt -> RngIntElt
                  Reduction(D) : DivFunElt -> DivFunElt, RngIntElt, DivFunElt, FldFunElt
                  GapNumbers(D, P) : DivFunElt, PlcFunElt -> SeqEnum[RngIntElt]
                  GapNumbers(D) : DivFunElt -> SeqEnum[RngIntElt]
                  Example FldFunG_divisors (H42E36)
                  Example FldFunG_AlgReln1 (H42E37)
                  Example FldFunG_AlgReln2 (H42E38)
                  RamificationDivisor(D) : DivFunElt -> DivFunElt
                  WeierstrassPlaces(D) : DivFunElt -> [PlcFunElt]
                  IsWeierstrassPlace(D, P) : DivFunElt, PlcFunElt -> BoolElt
                  WronskianOrders(D) : DivFunElt -> [RngIntElt]
                  ComplementaryDivisor(D) : DivFunElt -> DivFunElt
                  DifferentialBasis(D) : DivFunElt -> [DiffFunElt]
                  DifferentialSpace(D) : DivFunElt -> ModFld, Map
                  Parametrization(F, D) : FldFun, DivFunElt -> FldFunElt, [FldFunRatUElt]

      Functions related to Divisor Class Groups of Global Function Fields
            ClassGroupGenerationBound(q, g) : RngIntElt, RngIntElt -> RngIntElt
            ClassGroupGenerationBound(F) : FldFunG -> RngIntElt
            ClassNumberApproximation(F, e) : FldFunG, FldReElt -> FldReElt
            ClassNumberApproximationBound(q, g, e) : RngIntElt, RngIntElt, RngIntElt, -> RngIntElt
            ClassGroup(F : parameters) : FldFun -> GrpAb, Map, Map
            ClassGroupAbelianInvariants(F : parameters) : FldFun -> SeqEnum
            ClassNumber(F) : FldFun -> RngIntElt
            Example FldFunG_divisors-class (H42E39)
            GlobalUnitGroup(F) : FldFun -> GrpAb, Map
            IsGlobalUnit(a) : FldFunElt -> BoolElt
            IsGlobalUnitWithPreimage(a) : FldFunElt -> BoolElt, GrpAbElt
            PrincipalDivisorMap(F) : FldFunG -> Map
            ClassGroupExactSequence(F) : FldFunG -> Map, Map, Map
            SUnitGroup(S) : SetEnum[PlcFunElt] -> GrpAb, Map
            IsSUnit(a, S) : FldFunElt, SetEnum[PlcFunElt] -> BoolElt
            IsSUnitWithPreimage(a, S) : FldFunElt, SetEnum[PlcFunElt] -> BoolElt, GrpAbElt
            SRegulator(S) : SetEnum[PlcFunElt] -> RngIntElt
            SPrincipalDivisorMap(S) : SetEnum[PlcFunElt] -> Map
            IsSPrincipal(D, S) : DivFunElt, SetEnum[PlcFunElt] -> BoolElt, FldFunElt
            SClassGroup(S) : SetEnum[PlcFunElt] -> GrpAb, Map, Map
            SClassGroupExactSequence(S) : SetEnum[PlcFunElt] -> Map, Map, Map
            SClassGroupAbelianInvariants(S) : SetEnum[PlcFunElt] -> SeqEnum
            SClassNumber(S) : SetEnum[PlcFunElt] -> RngIntElt
            ClassGroupPRank(F) : FldFunG -> RngIntElt
            HasseWittInvariant(F) : FldFunG -> RngIntElt
            TateLichtenbaumPairing(D1, D2, m) : DivFunElt, DivFunElt, RngIntElt -> RngElt
            Example FldFunG_tate (H42E40)

 
Differentials

      Creation of Structures
            DifferentialSpace(F) : FldFunG -> DiffFun

      Creation of Elements
            Differential(a) : FldFunGElt -> DiffFunElt
            Identity(D) : DiffFun -> DiffFunElt
            IsCanonical(D) : DivFunElt -> BoolElt, DiffFunElt

      Related Structures
            FunctionField(D) : DiffFun -> FldFun
            FunctionField(d) : DiffFunElt -> FldFun

      Subspaces
            SpaceOfDifferentialsFirstKind(F) : FldFunG -> ModFld, Map
            BasisOfDifferentialsFirstKind(F) : FldFunG -> SeqEnum[DiffFunElt]
            DifferentialBasis(D) : DivFunElt -> [DiffFunElt]
            DifferentialSpace(D) : DivFunElt -> ModFld, Map
            Example FldFunG_div_diff (H42E41)

      Structure Predicates
            D1 eq D2 : DiffFun, DiffFun -> BoolElt

      Operations on Elements

            Arithmetic Operators
                  r * x : RngElt, DiffFunElt -> DiffFunElt

            Equality and Membership
                  x eq y : DiffFunElt, DiffFunElt -> BoolElt
                  x in D : Any, DiffFun -> BoolElt

            Predicates on Elements
                  IsExact(d) : DiffFunElt -> BoolElt, FldFunGElt
                  IsZero(d) : DiffFunElt -> BoolElt

            Functions on Elements
                  Valuation(d, P) : DiffFunElt, PlcFunElt -> RngIntElt
                  Divisor(d) : DiffFunElt -> DivFunElt
                  Residue(d, P) : DiffFunElt, PlcFunElt -> RngElt
                  Example FldFunG_diff-fun (H42E42)
                  Module(L, R) : SeqEnum[ DiffFunElt ], Rng -> Mod, Map, SeqEnum[ ModElt ]
                  Relations(L, R) : SeqEnum[ DiffFunElt ], Rng -> ModTupRng
                  Example FldFunG_module-diff (H42E43)
                  Cartier(b) : DiffFunElt -> DiffFunElt

            Other
                  CartierRepresentation(F) : FldFunG -> AlgMatElt, SeqEnum[DiffFunElt]
                  Example FldFunG_diff-cart (H42E44)

 
Weil Descent
      WeilDescent(E,k) : FldFun, FldFin -> FldFunG, Map
      ArtinSchreierExtension(c,a,b) : FldFin, FldFin, FldFin -> FldFun
      WeilDescentDegree(E,k) : FldFun, FldFin -> RngIntElt
      WeilDescentGenus(E,k) : FldFun, FldFin -> RngIntElt
      MultiplyFrobenius(b,f,F) : RngElt, RngUPolElt, Map -> RngElt
      Example FldFunG_ghs-descent (H42E45)

 
Function Field Database

      Creation
            FunctionFieldDatabase(q, d) : RngIntElt, RngIntElt -> DB
            sub< D | : parameters> : DB -> DB

      Access
            BaseField(D) : DB -> FldFin
            Degree(D) : DB -> RngIntElt
            # D : DB -> RngIntElt
            FunctionFields(D) : DB -> [ FldFunG ]
            Example FldFunG_alffdb-basic1 (H42E46)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012