Introduction

 
Creation of Quaternion Algebras
      QuaternionAlgebra< K | a, b > : Rng, RngElt, RngElt -> AlgQuat
      AssignNames(~A, S) : AlgQuat, [MonStgElt] ->
      Example AlgQuat_Quaternion_Constructor (H95E1)
      Example AlgQuat_Quaternion_Constructor_char2 (H95E2)
      QuaternionAlgebra(N) : RngIntElt -> AlgQuat
      QuaternionAlgebra(N) : RngUPolElt -> AlgQuat
      QuaternionAlgebra(I) : RngOrdIdl -> AlgQuat
      QuaternionAlgebra(I, S) : RngOrdIdl, [PlcNumElt] -> AlgQuat
      QuaternionAlgebra(S) : [PlcNumElt] -> AlgQuat
      Example AlgQuat_Quaternion_Constructor_Over_NumberField (H95E3)
      QuaternionAlgebra(D1, D2, T) : RngIntElt, RngIntElt, RngIntElt -> AlgQuat
      Example AlgQuat_Quaternion_Constructor_over_Rationals (H95E4)

 
Creation of Quaternion Orders

      Creation of Orders from Elements
            QuaternionOrder(S) : [AlgQuatElt] -> AlgQuatOrd
            QuaternionOrder(R, S) : Rng, [AlgQuatElt] -> AlgQuatOrd
            Example AlgQuat_Quaternion_Orders_over_Q_FqX (H95E5)

      Creation of Maximal Orders
            MaximalOrder(A) : AlgQuat[FldRat] -> AlgQuatOrd
            Example AlgQuat_Quaternion_MaximalOrder (H95E6)
            MaximalOrder(O) : AlgQuatOrd -> AlgQuat
            pMaximalOrder(O, p) : AlgQuatOrd, RngElt -> AlgQuatOrd, RngIntElt
            TameOrder(A) : AlgQuat[FldAlg] -> AlgAssVOrd

      Creation of Orders with given Discriminant
            Order(O, N) : AlgQuatOrd, RngElt -> AlgQuatOrd
            Order(O, N) : AlgAssVOrd, RngOrdIdl -> AlgAssVOrd
            GorensteinClosure(O) : AlgAssVOrd -> AlgAssVOrd, .
            Example AlgQuat_Quaternion_Orders (H95E7)

      Creation of Orders with given Discriminant over the Integers
            QuaternionOrder(A, M) : AlgQuat[FldRat], RngIntElt -> AlgQuatOrd
            QuaternionOrder(N) : RngIntElt -> AlgQuatOrd
            QuaternionOrder(D1, D2, T) : RngIntElt, RngIntElt, RngIntElt -> AlgQuat
            Example AlgQuat_Quaternion_Orders_over_the_Integers (H95E8)

 
Elements of Quaternion Algebras

      Creation of Elements
            A ! 0 : AlgQuat, RngIntElt -> AlgQuatElt
            A ! 1 : AlgQuat, RngIntElt -> AlgQuatElt
            A . i : AlgQuat, RngIntElt -> AlgQuatElt
            A ! x : AlgQuat, Any -> AlgQuatElt

      Arithmetic of Elements
            x + y : AlgQuatElt, AlgQuatElt -> AlgQuatElt
            x - y : AlgQuatElt, AlgQuatElt -> AlgQuatElt
            x * y : AlgQuatElt, AlgQuatElt -> AlgQuatElt
            x / y : AlgQuatElt, AlgQuatElt -> AlgQuatElt
            x eq y : AlgQuatElt, AlgQuatElt -> BoolElt
            x ne y : AlgQuatElt, AlgQuatElt -> BoolElt
            x in A : AlgQuatElt, AlgQuat -> BoolElt
            x notin A : AlgQuatElt, AlgQuat -> BoolElt
            Conjugate(x) : AlgQuatElt -> AlgQuatElt
            ElementToSequence(x) : AlgQuatElt -> SeqEnum
            Norm(x) : AlgQuatElt -> FldElt
            Trace(x) : AlgQuatElt -> FldElt
            CharacteristicPolynomial(x) : AlgQuatElt -> RngUPolElt
            MinimalPolynomial(x) : AlgQuatElt -> RngUPolElt
            Example AlgQuat_Element_Arithmetic (H95E9)

 
Attributes of Quaternion Algebras
      BaseField(A) : AlgQuat -> Fld
      Basis(A) : AlgQuat -> SeqEnum
      RamifiedPrimes(A) : AlgQuat -> SeqEnum
      Example AlgQuat_Ramified_Primes (H95E10)
      RamifiedPlaces(A) : AlgQuat -> SeqEnum, SeqEnum
      Example AlgQuat_Ramified_Primes_FqX (H95E11)
      Discriminant(A) : AlgQuat[FldRat] -> RngIntElt
      StandardForm(A) : AlgQuat -> RngElt, RngElt, AlgQuat, Map

 
Hilbert Symbols and Embeddings
      HilbertSymbol(a, b, p) : FldRatElt, FldRatElt, RngIntElt -> RngIntElt
      IsRamified(p, A) : RngElt, AlgQuat -> BoolElt
      Example AlgQuat_Hilbert_Symbols (H95E12)
      pMatrixRing(A, p) : AlgQuat, RngOrdIdl -> AlgMat, Map, Map
      IsSplittingField(K, A) : Fld, AlgQuat -> BoolElt, AlgQuatElt, Map
      Embed(K, A) : Fld, AlgQuat -> AlgQuatElt, Map
      Embed(Oc, O) : RngOrd, AlgAssVOrd -> AlgAssVOrdElt, Map
      Example AlgQuat_Embed (H95E13)

 
Predicates on Algebras
      IsDefinite(A) : AlgQuat -> BoolElt

 
Recognition Functions
      IsMatrixRing(A) : AlgQuat -> BoolElt, AlgMat, Map
      MatrixRing(A, eps) : AlgQuat, AlgQuatElt -> AlgMat, Map
      Example AlgQuat_Quaternion_MatrixRing (H95E14)
      IsQuaternionAlgebra(B) : AlgAss -> BoolElt, AlgQuat, Map
      Example AlgQuat_Quaternion_IsQuaternionAlgebra (H95E15)
      MatrixRepresentation(A) : AlgQuat -> Map

 
Attributes of Orders
      Algebra(S) : AlgQuatOrd -> AlgQuat
      BasisMatrix(S) : AlgQuatOrd -> AlgMatElt
      Discriminant(S) : AlgQuatOrd -> RngElt
      FactoredDiscriminant(S) : AlgQuatOrd -> SeqEnum
      Conductor(S) : AlgQuatOrd -> RngElt
      Normalizer(S) : AlgAssVOrd -> Grp, Map

 
Predicates of Orders
      IsMaximal(O) : AlgAssVOrd -> BoolElt
      IspMaximal(O, p) : AlgAssVOrd, RngOrdIdl -> BoolElt
      IsEichler(O) : AlgAssVOrd -> BoolElt, AlgAssVOrd, AlgAssVOrd
      IsEichler(O, p) : AlgAssVOrd , RngOrdIdl -> BoolElt, AlgAssVOrd, AlgAssVOrd
      EichlerInvariant(O, p) : AlgAssVOrd , RngOrdIdl -> RngIntElt
      IsHereditary(O) : AlgAssVOrd -> BoolElt
      IsHereditary(O, p) : AlgAssVOrd , RngOrdIdl -> BoolElt
      IsGorenstein(O) : AlgAssVOrd -> BoolElt, .
      IsGorenstein(O, p) : AlgAssVOrd , RngOrdIdl -> BoolElt, RngIntElt
      IsBass(O) : AlgAssVOrd -> BoolElt
      IsBass(O, p) : AlgAssVOrd , RngOrdIdl -> BoolElt
      IsSameType(O1, O2) : AlgAssVOrd , AlgAssVOrd -> BoolElt

 
Operations with Orders
      O1 meet O2 : AlgQuatOrd[RngInt], AlgQuatOrd[RngInt] -> AlgQuatOrd
      O ^ x : AlgQuatOrd, AlgQuatElt -> AlgQuatOrd

 
Ideal Theory of Orders

      Creation and Access Functions
            LeftIdeal(S, X) : AlgQuatOrd, [AlgQuatElt] -> AlgQuatOrdIdl
            PrimeIdeal(S, p) : AlgQuatOrd, RngElt -> AlgQuatOrdIdl
            CommutatorIdeal(S) : AlgQuatOrd -> AlgQuatOrdIdl
            MaximalLeftIdeals(O, p) : AlgQuatOrd, RngElt -> [AlgQuatOrdIdl]
            Example AlgQuat_Elementary_Ideals (H95E16)
            Example AlgQuat_Ideal_Bases (H95E17)
            LeftOrder(I) : AlgQuatOrdIdl -> AlgQuatOrd
            RightOrder(I) : AlgQuatOrdIdl -> AlgQuatOrd
            Example AlgQuat_Left_Right_Quaternion_Ordre (H95E18)

      Enumeration of Ideal Classes
            Mass(S) : AlgAssVOrd -> FldRatElt
            LeftIdealClasses(S) : AlgQuatOrd -> [AlgQuatOrdIdl]
            TwoSidedIdealClasses(S) : AlgQuatOrd -> [AlgQuatOrdIdl]
            TwoSidedIdealClassGroup(S : Support) : AlgAssVOrd -> GrpAb, Map
            ConjugacyClasses(S) : AlgAssVOrd -> SeqEnum
            Example AlgQuat_Ideal_Enumeration (H95E19)
            Example AlgQuat_Ideal_Enumeration (H95E20)
            Example AlgQuat_Ideal_Enumeration (H95E21)

      Operations on Ideals
            I * J : AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl
            I meet J : AlgQuatOrdIdl, AlgQuatOrdIdl -> AlgQuatOrdIdl
            Conjugate(I) : AlgQuatOrdIdl -> AlgQuatOrdIdl
            Norm(I) : AlgQuatOrdIdl -> RngElt
            Factorization(I) : AlgQuatOrdIdl -> SeqEnum

 
Norm Spaces and Basis Reduction
      NormSpace(A) : AlgQuat -> ModTupFld, Map
      NormSpace(S) : AlgQuatOrd -> ModTupRng, Map
      GramMatrix(S) : AlgQuatOrd -> AlgMatElt
      ReducedGramMatrix(S) : AlgQuatOrd[RngInt] -> AlgMatElt
      ReducedBasis(S) : AlgQuatOrd[RngInt] -> SeqEnum
      Example AlgQuat_Basis_Reduction (H95E22)
      ReducedGramMatrix(S) : AlgQuatOrd[RngUPol] -> AlgMatElt, SeqEnum
      ReducedBasis(O) : AlgAssVOrd[RngOrd] -> [AlgAssVElt]
      OptimizedRepresentation(O) : AlgAssVOrd -> AlgQuat, Map
      OptimizedRepresentation(A) : AlgQuat -> AlgQuat, Map
      Enumerate(O, A, B) : AlgQuatOrd[RngInt], RngIntElt, RngIntElt -> [AlgQuatOrdElt]
      Enumerate(O, A, B) : AlgAssVOrd[RngOrd], RngElt, RngElt -> [AlgAssVOrdElt]

 
Isomorphisms

      Isomorphisms of Algebras
            IsIsomorphic(A, B) : AlgQuat, AlgQuat -> BoolElt, Map

      Isomorphisms of Orders
            IsIsomorphic(S, T) : AlgQuatOrd, AlgQuatOrd -> BoolElt, Map, AlgQuatElt
            Isomorphism(S, T) : AlgQuatOrd, AlgQuatOrd -> Map

      Isomorphisms of Ideals
            IsIsomorphic(I, J) : AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt, AlgAssVElt
            IsPrincipal(I) : AlgAssVOrdIdl -> BoolElt, AlgQuatElt
            IsLeftIsomorphic(I, J) : AlgQuatOrdIdl, AlgQuatOrdIdl -> BoolElt, Map, AlgQuatElt
            IsLeftIsomorphic(I, J) : AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> BoolElt, AlgQuatElt
            LeftIsomorphism(I, J) : AlgQuatOrdIdl, AlgQuatOrdIdl -> Map, AlgQuatElt
            RightIsomorphism(I, J) : AlgQuatOrdIdl, AlgQuatOrdIdl -> Map, AlgQuatElt

      Examples
            Example AlgQuat_Isomorphism_algebras (H95E23)
            Example AlgQuat_Isomorphism_example (H95E24)
            Example AlgQuat_Left_Right_Isomorphisms (H95E25)
            Example AlgQuat_Left_Right_Isomorphisms_Number_Field (H95E26)

 
Units and Unit Groups
      NormOneGroup(S) : AlgAssVOrd -> GrpPerm, Map
      Units(S) : AlgQuatOrd -> SeqEnum
      MultiplicativeGroup(S) : AlgQuatOrd[RngInt] -> GrpPerm, Map
      Example AlgQuat_Unit_Group (H95E27)
      Example AlgQuat_Unit_Group_NumberRing (H95E28)

 
Bibliography

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