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ORDERS AND ALGEBRAIC FIELDS

 
Acknowledgements
 
Introduction
 
Creation Functions
      Creation of General Algebraic Fields
      Creation of Orders and Fields from Orders
      Maximal Orders
            Orders and Ideals
      Creation of Elements
      Creation of Homomorphisms
 
Special Options
 
Structure Operations
      General Functions
      Related Structures
      Representing Fields as Vector Spaces
      Invariants
      Basis Representation
      Ring Predicates
      Order Predicates
      Field Predicates
      Setting Properties of Orders
 
Element Operations
      Parent and Category
      Arithmetic
      Equality and Membership
      Predicates on Elements
      Finding Special Elements
      Real and Complex Valued Functions
      Norm, Trace, and Minimal Polynomial
      Other Functions
 
Ideal Class Groups
      Setting the Class Group Bounds Globally
 
Unit Groups
 
Solving Equations
      Norm Equations
      Thue Equations
      Unit Equations
      Index Form Equations
 
Ideals and Quotients
      Creation of Ideals in Orders
      Invariants
      Basis Representation
      Two--Element Presentations
      Predicates on Ideals
      Ideal Arithmetic
      Roots of Ideals
      Factorization and Primes
      Other Ideal Operations
      Quotient Rings
            Operations on Quotient Rings
            Elements of Quotients
            Reconstruction
 
Places and Divisors
      Creation of Structures
      Operations on Structures
      Creation of Elements
      Arithmetic with Places and Divisors
      Other Functions for Places and Divisors
 
Bibliography







DETAILS

 
Introduction

 
Creation Functions

      Creation of General Algebraic Fields
            NumberField(f) : RngUPolElt -> FldNum
            RationalsAsNumberField() : FldRat -> FldNum
            NumberField(s) : [ RngUPolElt ] -> FldNum
            ext< F | s1, ..., sn > : FldAlg, RngUPolElt, ..., RngUPolElt -> FldAlg
            RadicalExtension(F, d, a) : Rng, RngIntElt, RngElt -> FldAlg
            SplittingField(F) : FldAlg -> FldAlg, SeqEnum
            SplittingField(f) : RngUPolElt -> FldAlg
            SplittingField(L) : [RngUPolElt] -> FldNum, [FldNumElt]
            sub< F | e1, ..., en > : FldAlg, FldAlgElt, ..., FldAlgElt -> FldAlg, Map
            MergeFields(F, L) : FldAlg, FldAlg -> SeqEnum
            Compositum(K, L) : FldAlg, FldAlg -> FldAlg
            Compositum(K, A) : FldAlg, FldAb -> FldAlg
            OptimizedRepresentation(F) : FldAlg -> FldAlg, map
            Example RngOrd_opt-rep-ord (H37E1)

      Creation of Orders and Fields from Orders
            EquationOrder(f) : RngUPolElt -> RngOrd
            EquationOrder(K) : FldNum -> RngOrd
            SubOrder(O) : RngOrd -> RngOrd
            EquationOrder(O) : RngOrd -> RngOrd
            Integers(O) : RngOrd -> RngOrd
            Example RngOrd_Orders (H37E2)
            sub< O | a1, ..., ar > : RngOrd, RngOrdElt, ..., RngOrdElt -> RngOrd
            ext< O | a1, ..., ar > : RngOrd, RngOrdElt, ..., RngOrdElt -> RngOrd
            ext< Z | f > : RngInt, RngUPolElt -> RngOrd
            FieldOfFractions(O) : RngOrd -> FldOrd
            Order(F) : FldOrd -> RngOrd
            NumberField(O) : RngOrd -> FldNum
            NumberField(F) : FldOrd -> FldNum
            Example RngOrd_fractions (H37E3)
            OptimizedRepresentation(O) : RngOrd -> BoolElt, RngOrd, Map
            O + P : RngOrd, RngOrd -> RngOrd
            O meet P : RngOrd, RngOrd -> RngOrd
            AsExtensionOf(O, P) : RngOrd, RngOrd -> RngOrd
            Order(O, T, d) : RngOrd, AlgMatElt, RngIntElt -> RngOrd
            Order(O, M) : RngOrd, ModDed -> RngOrd
            Order( [ e1, ... en ] ): [FldAlgElt] -> RngOrd

      Maximal Orders
            MaximalOrder(O) : RngOrd -> RngOrd
            MaximalOrder(F) : FldAlg -> RngOrd
            MaximalOrder(f) : RngUPolElt -> RngOrd
            Example RngOrd_max_order (H37E4)

            Orders and Ideals
                  pMaximalOrder(O, p) : RngOrd, RngIntElt -> RngOrd
                  pRadical(O, p) : RngOrd, RngIntElt -> RngOrdIdl
                  MultiplicatorRing(I) : RngOrdFracIdl -> Rng
                  Example RngOrd_Round2 (H37E5)

      Creation of Elements
            F ! a : FldAlg, RngElt -> FldAlgElt
            F ! [a0, a1, ..., am - 1] : FldAlg, [RngElt] -> FldAlgElt
            O ! a : RngOrd, RngElt -> RngOrdElt
            O ! [a0, a1, ..., am - 1] : RngOrd, [ RngElt ] -> RngOrdElt
            Random(F, m) : FldAlg, RngIntElt -> FldAlgElt
            Random(I, m) : RngOrdFracIdl, RngIntElt -> FldOrdElt
            Example RngOrd_Elements (H37E6)

      Creation of Homomorphisms
            hom< F -> R | r > : FldAlg, Rng, RngElt -> Map
            hom< O -> R | r > : RngOrd, Rng, RngElt -> Map
            Example RngOrd_Homomorphisms (H37E7)
            hom< O -> R | b1, ..., bn > : RngFunOrd, Rng, RngElt, ..., RngElt -> Map
            IsRingHomomorphism(m) : Map -> BoolElt

 
Special Options
      SetVerbose(s, n) : MonStgElt, RngIntElt ->
      SetKantPrinting(f) : BoolElt -> BoolElt
      SetKantPrecision(n) : RngIntElt ->

 
Structure Operations

      General Functions
            AssignNames(~K, s) : FldNum, [ MonStgElt ] ->
            Name(K, i) : FldNum, RngIntElt -> FldNumElt
            AssignNames(~F, s) : FldOrd, [ MonStgElt ] ->
            F . i : FldOrd, RngIntElt -> FldOrdElt
            O . i : RngOrd, RngIntElt -> RngOrdElt

      Related Structures
            GroundField(F) : FldAlg -> Fld
            BaseRing(O) : RngOrd -> Rng
            AbsoluteField(F) : FldAlg -> FldAlg
            AbsoluteOrder(O) : RngOrd -> RngOrd
            SimpleExtension(F) : FldAlg -> FldAlg
            RelativeField(F, L) : FldAlg, FldAlg -> FldAlg
            Example RngOrd_Compositum (H37E8)
            Simplify(O) : RngOrd -> RngOrd
            LLL(O) : RngOrd -> RngOrd, AlgMatElt
            Example RngOrd_lll (H37E9)
            Embed(F, L, a) : FldAlg, FldAlg, FldAlgElt ->
            Embed(F, L, a) : FldAlg, FldAlg, [FldAlgElt] ->
            EmbeddingMap(F, L): FldAlg, FldAlg -> Map
            Example RngOrd_em (H37E10)
            Lattice(O) : RngOrd -> Lat, Map
            MinkowskiSpace(F) : FldAlg -> Lat, Map
            Completion(K, P) : FldAlg, RngOrdIdl -> FldLoc, Map
            Completion(K, P) : FldAlg, PlcNumElt -> FldLoc, Map
            LocalRing(P, prec) : RngOrdIdl, RngIntElt -> RngLoc, Map

      Representing Fields as Vector Spaces
            Algebra(K, J) : FldAlg, Fld -> AlgAss, Map
            VectorSpace(K, J) : FldAlg, Fld -> ModTupFld, Map
            Example RngOrd_vector_space_eg (H37E11)

      Invariants
            Degree(O) : RngOrd -> RngIntElt
            AbsoluteDegree(O) : RngOrd -> RngIntElt
            Discriminant(O) : RngOrd -> RngIntElt
            AbsoluteDiscriminant(O) : RngOrd -> RngIntElt
            AbsoluteDiscriminant(K) : FldAlg -> FldRatElt
            ReducedDiscriminant(O) : RngOrd -> RngIntElt
            Regulator(O: parameters) : RngOrd -> FldPrElt
            RegulatorLowerBound(O) : RngOrd -> FldPrElt
            Signature(O) : RngOrd -> RngIntElt, RngIntElt
            UnitRank(O) : RngOrd -> RngIntElt
            Index(O, S) : RngOrd, RngOrd -> RngIntElt
            DefiningPolynomial(F) : FldAlg -> RngUPolElt
            Zeroes(O, n) : RngOrd, RngIntElt -> [ FldPrElt ]
            Example RngOrd_zero (H37E12)
            Different(O) : RngOrd -> RngOrdIdl
            Conductor(O) : RngOrd -> RngOrdIdl

      Basis Representation
            Basis(O) : RngOrd -> [ FldOrdElt ]
            IntegralBasis(F) : FldAlg -> [ FldAlgElt ]
            Example RngOrd_basis-ring (H37E13)
            AbsoluteBasis(K) : FldAlg -> [FldAlgElt]
            BasisMatrix(O) : RngOrd -> AlgMatElt
            TransformationMatrix(O, P) : RngOrd, RngOrd -> AlgMatElt, RngIntElt
            CoefficientIdeals(O) : RngOrd -> [RngOrdFracIdl]
            Example RngOrd_Bases (H37E14)
            MultiplicationTable(O) : RngOrd -> [AlgMatElt]
            TraceMatrix(O) : RngOrd -> AlgMatElt
            Example RngOrd_MultiplicationTable (H37E15)

      Ring Predicates
            N eq O : RngOrd, RngOrd -> BoolElt
            F eq L : FldAlg, FldAlg -> BoolElt
            IsEuclideanDomain(F) : FldAlg -> BoolElt
            IsSimple(F) : FldAlg -> BoolElt
            IsPrincipalIdealRing(F) : FldAlg -> BoolElt
            IsPrincipalIdealRing(O) : RngOrd -> BoolElt
            HasComplexConjugate(K) : FldAlg -> BoolElt, Map
            ComplexConjugate(x) : FldAlgElt -> FldAlgElt

      Order Predicates
            IsEquationOrder(O) : RngOrd -> BoolElt
            IsMaximal(O) : RngOrd -> BoolElt
            IsAbsoluteOrder(O) : RngOrd -> BoolElt
            IsWildlyRamified(O) : RngOrd -> BoolElt
            IsTamelyRamified(O) : RngOrd -> BoolElt
            IsUnramified(O) : RngOrd -> BoolElt

      Field Predicates
            IsIsomorphic(F, L) : FldAlg, FldAlg -> BoolElt, Map
            IsSubfield(F, L) : FldAlg, FldAlg -> BoolElt, Map
            IsNormal(F) : FldAlg -> BoolElt
            IsAbelian(F) : FldAlg -> BoolElt
            IsCyclic(F) : FldAlg -> BoolElt
            IsAbsoluteField(K) : FldAlg -> BoolElt
            IsWildlyRamified(K) : FldAlg -> BoolElt
            IsTamelyRamified(K) : FldAlg -> BoolElt
            IsUnramified(K) : FldAlg -> BoolElt
            IsQuadratic(K) : FldAlg -> BoolElt, FldQuad
            IsTotallyReal(K) : FldAlg -> BoolElt

      Setting Properties of Orders
            SetOrderMaximal(O, b) : RngOrd, BoolElt ->
            SetOrderTorsionUnit(O, e, r) : RngOrd, RngOrdElt, RngIntElt ->
            SetOrderUnitsAreFundamental(O) : RngOrd ->

 
Element Operations

      Parent and Category

      Arithmetic
            w div v : RngOrdElt, RngOrdElt -> RngOrdElt
            Modexp(a, n, m) : RngOrdElt, RngIntElt, RngIntElt -> RngOrdElt
            Sqrt(a) : RngOrdElt -> RngOrdElt
            Root(a, n) : RngOrdElt, RngIntElt -> RngOrdElt
            IsPower(a, k) : FldAlgElt, RngIntElt -> BoolElt, FldAlgElt
            Denominator(a) : FldAlgElt -> RngIntElt
            Numerator(a) : FldAlgElt -> RngIntElt
            Qround(E, M): FldAlgElt, RngIntElt -> FldAlgElt

      Equality and Membership

      Predicates on Elements
            IsIntegral(a) : FldAlgElt -> BoolElt
            IsPrimitive(a) : FldAlgElt -> BoolElt
            IsTorsionUnit(w) : RngOrdElt -> BoolElt
            IsPower(w, n) : RngOrdElt, RngIntElt -> BoolElt, RngOrdElt
            IsTotallyPositive(a) : RngOrdElt -> BoolElt

      Finding Special Elements
            K . 1 : FldNum -> FldNumElt
            PrimitiveElement(K) : FldNum -> FldNumElt
            Generators(K): FldAlg -> [FldAlgElt]
            Generators(K, k) : FldAlg, FldAlg -> [FldAlgElt]
            PrimitiveElement(O) : RngOrd -> RngOrdElt

      Real and Complex Valued Functions
            AbsoluteValues(a) : FldAlgElt -> [FldPrElt]
            AbsoluteLogarithmicHeight(a) : FldAlgElt -> FldPrElt
            Conjugates(a) : FldAlgElt -> [ FldComElt ]
            Conjugate(a, k) : FldAlgElt, RngIntElt -> FldPrElt
            Conjugate(a, l) : FldAlgElt, [RngIntElt] -> FldReElt
            Length(a) : FldAlgElt -> FldPrElt
            Logs(a) : FldAlgElt -> [FldPrElt]
            CoefficientHeight(E) : RngOrdElt -> RngIntElt
            CoefficientLength(E) : RngOrdElt -> RngIntElt
            Example RngOrd_Discriminant (H37E16)

      Norm, Trace, and Minimal Polynomial
            Norm(a) : FldAlgElt -> FldAlgElt
            AbsoluteNorm(a) : FldAlgElt -> FldRatElt
            Trace(a) : FldAlgElt -> FldAlgElt
            AbsoluteTrace(a) : FldAlgElt -> FldRatElt
            CharacteristicPolynomial(a) : FldAlgElt -> RngUPolElt
            AbsoluteCharacteristicPolynomial(a) : FldAlgElt -> RngUPolElt
            MinimalPolynomial(a) : FldAlgElt -> RngUPolElt
            AbsoluteMinimalPolynomial(a) : FldAlgElt -> RngUPolElt
            RepresentationMatrix(a) : FldAlgElt -> AlgMatElt
            AbsoluteRepresentationMatrix(a) : FldAlgElt -> AlgMatElt
            Example RngOrd_NormsEtc (H37E17)

      Other Functions
            ElementToSequence(a) : FldAlgElt -> [ FldAlgElt ]
            Eltseq(E, k) : FldAlgElt, FldAlg -> [RngElt]
            Flat(e) : FldAlgElt -> [ FldRatElt]
            a[i] : FldAlgElt, RngIntElt -> FldRatElt
            ProductRepresentation(a) : RngOrdElt -> [ RngOrdElt ], [ RngIntElt ]
            ProductRepresentation(P, E) : [ FldAlgElt ], [ RngIntElt ] -> FldAlgElt
            Valuation(w, I) : RngOrdElt, RngOrdIdl -> RngIntElt
            Decomposition(a) : RngOrdElt -> SeqEnum[<RngOrdIdl, RngIntElt>]
            Divisors(a) : RngOrdElt -> SeqEnum[RngOrdElt]
            Index(a) : RngOrdElt -> RngIntElt
            Different(a) : RngOrdElt -> RngOrdElt

 
Ideal Class Groups
      DegreeOnePrimeIdeals(O, B) : RngOrd, RngIntElt -> [ RngOrdIdl ]
      ClassGroup(O: parameters) : RngOrd -> GrpAb, Map
      RingClassGroup(O) : RngOrd -> GrpAb, Map
      ConditionalClassGroup(O) : RngOrd -> GrpAb, Map
      ClassGroupPrimeRepresentatives(O, I) : RngOrd, RngOrdIdl -> Map
      ClassNumber(O: parameters) : RngOrd -> RngIntElt
      BachBound(K) : FldNum -> RngIntElt
      MinkowskiBound(K) : FldNum -> RngIntElt
      FactorBasis(K, B) : FldNum, RngIntElt -> [ RngOrdIdl ]
      FactorBasis(O) : RngOrd -> [ RngOrdIdl ], Integer
      RelationMatrix(K, B) : FldNum, RngIntElt -> ModHomElt
      RelationMatrix(O) : RngOrd -> ModHomElt
      Relations(O) : RngOrd -> ModHomElt
      ClassGroupCyclicFactorGenerators(O) : RngOrd -> ModHomElt
      Example RngOrd_ClassGroup (H37E18)
      Example RngOrd_class-group-sieve (H37E19)
      FactorBasisCreate(O,B): RngOrd, RngIntElt -> SeqEnum
      EulerProduct(O, B) : RngOrd, RngIntElt -> FldReElt
      AddRelation(E) : RngOrdElt -> BoolElt
      EvaluateClassGroup(O) : RngOrd -> BoolElt
      CompleteClassGroup(O) : RngOrd ->
      FactorBasisVerify(O, L, U): RngOrd, RngIntElt, RngIntElt ->
      ClassGroupSetUseMemory(O, f) : RngOrd, BoolElt ->
      ClassGroupGetUseMemory(O) : RngOrd -> BoolElt

      Setting the Class Group Bounds Globally
            SetClassGroupBounds(n) : Any ->
            SetClassGroupBoundMaps(f1, f2) : Map, Map ->
            Example RngOrd_class-group-bounds (H37E20)

 
Unit Groups
      UnitGroup(O) : RngOrd -> GrpAb, Map
      UnitGroupAsSubgroup(O) : RngOrd -> GrpAb
      TorsionUnitGroup(O) : RngOrd -> GrpAb, Map
      IndependentUnits(O) : RngOrd -> GrpAb, Map
      pFundamentalUnits(O, p) : RngOrd, RngIntElt -> GrpAb, Map
      MergeUnits(K, a) : FldNum, FldNumElt -> BoolElt
      UnitRank(O) : RngOrd -> RngIntElt
      Example RngOrd_UnitGroup (H37E21)
      IsExceptionalUnit(u) : RngOrdElt -> BoolElt
      ExceptionalUnitOrbit(u) : RngOrdElt -> [ RngOrdElt ]
      ExceptionalUnits(O) : RngOrd -> [ RngOrdElt ]

 
Solving Equations

      Norm Equations
            NormEquation(O, m) : RngOrd, RngIntElt -> BoolElt, [ RngOrdElt ]
            NormEquation(F, m) : FldAlg, RngIntElt -> BoolElt, [ FldAlgElt ]
            NormEquation(m, N): RngElt, Map -> BoolElt, RngElt
            IntegralNormEquation(a, N, O) : RngElt, Map, RngOrd -> BoolElt, [RngOrdElt]
            SimNEQ(K, e, f) : FldNum, FldNumElt, FldNumElt -> BoolElt, [FldNumElt]
            Example RngOrd_norm-equation (H37E22)

      Thue Equations
            Thue(f) : RngUPolElt -> Thue
            Thue(O) : RngOrd -> Thue
            Evaluate(t, a, b) : Thue, RngIntElt, RngIntElt -> RngIntElt
            Solutions(t, a) : Thue, RngIntElt -> [ [ RngIntElt, RngIntElt ] ]
            Example RngOrd_thue (H37E23)

      Unit Equations
            UnitEquation(a, b, c) : FldNumElt, FldNumElt, FldNumElt -> [ ModHomElt ]
            Example RngOrd_uniteq (H37E24)

      Index Form Equations
            IndexFormEquation(O, k) : RngOrd, RngIntElt -> [ RngOrdElt ]
            Example RngOrd_index-form (H37E25)

 
Ideals and Quotients

      Creation of Ideals in Orders
            x * O : RngElt, RngOrd -> RngOrdFracIdl
            F !! I : RngOrd, RngInt -> RngOrdFracIdl
            ideal< O | a1, a2, ... , am > : RngOrd, RngElt, ..., RngElt -> RngOrdFracIdl
            Example RngOrd_Ideals (H37E26)

      Invariants
            Order(I) : RngOrdFracIdl -> RngOrd
            Denominator(I) : RngOrdFracIdl -> RngIntElt
            PrimitiveElement(I) : RngOrdIdl -> RngOrdElt
            Index(O, I) : RngOrd, RngOrdIdl -> RngIntElt
            Norm(I) : RngOrdIdl -> RngIntElt
            MinimalInteger(I) : RngOrdIdl -> RngElt
            Minimum(I) : RngOrdFracIdl -> RngElt
            AbsoluteNorm(I) : RngOrdIdl -> RngIntElt
            CoefficientHeight(I) : RngOrdIdl -> RngIntElt
            CoefficientLength(I) : RngOrdIdl -> RngIntElt
            RamificationIndex(I, p) : RngOrdIdl, RngIntElt -> RngIntElt
            RamificationDegree(I) : RngOrdIdl -> RngIntElt
            ResidueClassField(O, I) : RngOrd, RngOrdIdl -> FldFin, Map
            Degree(I) : RngOrdIdl -> RngIntElt
            Valuation(I, p) : RngOrdFracIdl, RngOrdIdl -> RngIntElt
            Content(I) : RngOrdFracIdl -> RngIntElt
            Example RngOrd_ideal-invar (H37E27)

      Basis Representation
            Basis(I) : RngOrdIdl -> [RngOrdElt]
            BasisMatrix(I) : RngOrdFracIdl -> MtrxSpcElt
            TransformationMatrix(I) : RngOrdFracIdl -> MtrxSpcElt, RngIntElt
            CoefficientIdeals(I) : RngOrdFracIdl -> [RngOrdFracIdl]
            Example RngOrd_ideal-basis (H37E28)
            Module(I) : RngOrdFracIdl -> ModDed, Map

      Two--Element Presentations
            Generators(I) : RngOrdIdl -> [ RngOrdElt ]
            TwoElement(I) : RngOrdFracIdl -> FldOrdElt, FldOrdElt
            TwoElementNormal(I) : RngOrdIdl -> RngOrdElt, RngOrdElt, RngIntElt
            Example RngOrd_ideal-two (H37E29)

      Predicates on Ideals
            IsIntegral(I) : RngOrdFracIdl -> BoolElt
            IsZero(I) : RngOrdFracIdl -> BoolElt
            IsOne(I) : RngOrdIdl -> BoolElt
            IsPrime(I) : RngOrdIdl -> BoolElt, RngOrdIdl
            IsPrincipal(I) : RngOrdFracIdl -> BoolElt, FldOrdElt
            IsRamified(P) : RngOrdIdl -> BoolElt
            IsRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
            IsTotallyRamified(P) : RngOrdIdl -> BoolElt
            IsTotallyRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
            IsTotallyRamified(K) : FldAlg -> BoolElt
            IsTotallyRamified(O) : RngOrd -> BoolElt
            IsWildlyRamified(P) : RngOrdIdl -> BoolElt
            IsWildlyRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
            IsTamelyRamified(P) : RngOrdIdl -> BoolElt
            IsTamelyRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
            IsUnramified(P) : RngOrdIdl -> BoolElt
            IsUnramified(P, O) : RngOrdIdl, RngOrd -> BoolElt
            IsInert(P) : RngOrdIdl -> BoolElt
            IsInert(P, O) : RngOrdIdl, RngOrd -> BoolElt
            IsSplit(P) : RngOrdIdl -> BoolElt
            IsSplit(P, O) : RngOrdIdl, RngOrd -> BoolElt
            IsTotallySplit(P) : RngOrdIdl -> BoolElt
            IsTotallySplit(P, O) : RngOrdIdl, RngOrd -> BoolElt

      Ideal Arithmetic
            I * J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
            x * I : RngElt, RngOrdFracIdl -> RngOrdFracIdl
            &* L : [RngOrdFracIdl] -> RngOrdFracIdl
            I / J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
            I div J : RngOrdIdl, RngOrdIdl -> RngOrdIdl
            I / x : RngOrdFracIdl, RngElt -> RngOrdFracIdl
            I + J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
            I ^ k : RngOrdFracIdl, RngIntElt -> RngOrdFracIdl
            I eq J : RngOrdFracIdl, RngOrdFracIdl -> BoolElt
            I subset J : RngOrdIdl, RngOrdIdl -> BoolElt
            E in I: RngOrdElt, RngOrdIdl -> BoolElt
            LCM(I, J) : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
            GCD(I, J) : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
            Content(M) : Mtrx -> RngOrdFracIdl
            I meet J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
            &meet S : [RngOrdFracIdl] -> RngOrdFracIdl
            I meet R : RngOrdFracIdl, Rng -> Any
            a mod I : RngOrdElt, RngOrdIdl -> RngOrdElt
            InverseMod(E, M) : RngOrdElt, RngIntElt -> RngOrdElt
            ColonIdeal(I, J) : RngOrdIdl, RngOrdIdl -> RngOrdIdl
            IntegralSplit(I) : RngOrdFracIdl -> RngOrdIdl, RngElt

      Roots of Ideals
            Root(I, k) : RngOrdFracIdl, RngIntElt -> RngOrdFracIdl
            IsPower(I, k) : RngOrdFracIdl, RngIntElt -> BoolElt, RngOrdFracIdl
            SquareRoot(I) : RngOrdFracIdl -> RngOrdFracIdl
            IsSquare(I) : RngOrdFracIdl -> BoolElt, RngOrdFracIdl

      Factorization and Primes
            Decomposition(O, p) : RngOrd, RngIntElt -> [<RngOrdIdl, RngIntElt>]
            DecompositionType(O, p) : RngOrd, RngIntElt -> [<RngIntElt, RngIntElt>]
            Factorization(I) : RngOrdFracIdl -> [<RngOrdIdl, RngIntElt>]
            Divisors(I) : RngOrdIdl -> [<RngOrdIdl, RngIntElt>]
            Support(I) : RngOrdFracIdl -> RngOrdIdl
            Support(L) : [RngOrdFracIdl] -> RngOrdIdl
            CoprimeBasis(L) : [RngOrdFracIdl] -> RngOrdIdl
            CoprimeBasisInsert(~L, I) : [RngOrdIdl], RngOrdFracIdl ->
            PowerProduct(B, E) : [RngOrdFracIdl], [RngIntElt] -> RngOrdFracIdl

      Other Ideal Operations
            ChineseRemainderTheorem(I1, I2, e1, e2) : RngOrdIdl, RngOrdIdl, RngOrdElt, RngOrdElt -> RngOrdElt
            CRT(I1, L1, e1, L2) : RngOrdIdl, [RngIntElt], RngOrdElt, [RngIntElt] -> RngOrdElt
            Idempotents(I, J) : RngOrdIdl, RngOrdIdl -> BoolElt, RngOrdElt, RngOrdElt
            CoprimeRepresentative(I, J) : RngOrdIdl, RngOrdIdl -> FldOrdElt
            ClassRepresentative(I) : RngOrdFracIdl -> RngOrdFracIdl
            Lattice(I) : RngOrdIdl -> Lat, Map
            Different(I) : RngOrdFracIdl -> RngOrdFracIdl
            Codifferent(I) : RngOrdFracIdl -> RngOrdFracIdl
            SUnitGroup(I) : RngOrdFracIdl -> GrpAb, Map
            Example RngOrd_S-Units (H37E30)
            SUnitAction(SU, Act, S) : Map, Map, SeqEnum[RngOrdIdl] -> Map
            SUnitAction(SU, Act, S) : Map, SeqEnum[Map], SeqEnum[RngOrdIdl] -> [Map]
            SUnitDiscLog(SU, x, S) : Map, FldAlgElt, SeqEnum[RngOrdIdl] -> GrpAbElt
            Example RngOrd_S-Units, advanced (H37E31)

      Quotient Rings

            Operations on Quotient Rings
                  quo< O | I > : RngOrd, RngOrdIdl -> RngOrdRes
                  UnitGroup(OQ) : RngOrdRes -> GrpAb, Map
                  Modulus(OQ) : RngOrdRes -> RngOrdIdl
                  Example RngOrd_quotient (H37E32)

            Elements of Quotients
                  OQ ! a : RngOrdRes, Elt -> RngOrdResElt
                  a mod I : RngOrdElt, RngOrdIdl -> RngOrdElt
                  IsZero(a) : RngOrdResElt -> BoolElt
                  IsOne(a) : RngOrdResElt -> BoolElt
                  IsMinusOne(a) : RngOrdResElt -> BoolElt
                  IsUnit(a) : RngOrdResElt -> BoolElt
                  Eltseq(a) : RngOrdResElt -> []

            Reconstruction
                  ReconstructionEnvironment(p, k) : RngOrdIdl, RngIntElt -> RngOrdRecoEnv
                  Reconstruct(x, R) : RngOrdElt, RngOrdRecoEnv -> RngOrdElt
                  ChangePrecision(~ R, k) : RngOrdRecoEnv, RngIntElt ->
                  Example RngOrd_order-reco (H37E33)

 
Places and Divisors

      Creation of Structures
            Places(K) : FldNum -> PlcNum

      Operations on Structures
            NumberField(P) : PlcNum -> FldNum

      Creation of Elements
            Place(I) : RngOrdIdl -> PlcNumElt
            Decomposition(K, p) : FldAlg, RngIntElt -> SeqEnum
            Decomposition(K, p) : FldNum, PlcNumElt -> SeqEnum
            Decomposition(m, p) : Map[FldRat, FldAlg], RngIntElt -> SeqEnum[<PlcNumElt, RngIntElt>]
            InfinitePlaces(K) : FldAlg -> SeqEnum
            Divisor(pl) : PlcNumElt -> DivNumElt
            Divisor(I) : RngOrdFracIdl -> DivNumElt
            Divisor(x) : FldNumElt -> DivNumElt
            RealPlaces(K) : FldRat -> [PlcNumElt]

      Arithmetic with Places and Divisors

      Other Functions for Places and Divisors
            Valuation(a, p) : FldNumElt, PlcNumElt -> RngElt
            Valuation(I, p) : RngOrdFracIdl , PlcNumElt -> RngElt
            Support(D) : DivNumElt -> SeqEnum, SeqEnum
            Ideal(D) : DivNumElt -> RngOrdIdl
            Evaluate(x, p) : FldOrdElt, PlcNumElt -> RngElt
            RealEmbeddings(a) : RngOrdElt -> []
            RealSigns(a) : RngOrdElt -> []
            IsReal(p) : PlcNumElt -> BoolElt
            IsComplex(p) : PlcNumElt -> BoolElt
            IsFinite(p) : PlcNumElt -> BoolElt
            IsInfinite(p) : PlcNumElt -> BoolElt, RngIntElt
            Extends(P, p) : PlcNumElt, PlcNumElt -> BoolElt
            InertiaDegree(P) : PlcNumElt -> RngIntElt
            Degree(D) : DivNumElt -> RngElt
            NumberField(P) : PlcNumElt -> FldNum
            ResidueClassField(P) : PlcNumElt -> Fld
            UniformizingElement(P) : PlcNumElt -> FldNumElt
            LocalDegree(P) : PlcNumElt -> RngIntElt
            RamificationIndex(P) : PlcNumElt -> RngIntElt
            DecompositionGroup(P) : PlcNumElt -> GrpPerm

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013