MAGMA Computational Algebra System

QUATERNION ALGEBRAS

DETAILS

 
Introduction

 
Creation of Quaternion Algebras
      QuaternionAlgebra< K | a, b > : Rng, RngElt, RngElt -> AlgQuat
      AssignNames(~A, S) : AlgQuat, [MonStgElt] ->
      Example AlgQuat_Quaternion_Constructor (H89E1)
      Example AlgQuat_Quaternion_Constructor_char2 (H89E2)
      QuaternionAlgebra(N) : RngIntElt -> AlgQuat
      QuaternionAlgebra(I) : RngOrdIdl -> AlgQuat
      QuaternionAlgebra(I, S) : RngOrdIdl, [PlcNumElt] -> AlgQuat
      QuaternionAlgebra(S) : [PlcNumElt] -> AlgQuat
      Example AlgQuat_Quaternion_Constructor_Over_NumberField (H89E3)
      QuaternionAlgebra(D1, D2, T) : RngIntElt, RngIntElt, RngIntElt -> AlgQuat
      Example AlgQuat_Quaternion_Constructor_over_Rationals (H89E4)

 
Creation of Quaternion Orders
      MaximalOrder(A) : AlgQuat[FldRat] -> AlgQuatOrd
      Example AlgQuat_Quaternion_MaximalOrder (H89E5)
      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 (H89E6)
      Order(O, N) : AlgAssVOrd, RngOrdIdl -> AlgAssVOrd
      QuaternionOrder(S) : [AlgQuatElt[FldRat]] -> AlgQuatOrd
      QuaternionOrder(R, S) : Rng, [AlgQuatElt] -> AlgQuatOrd
      Example AlgQuat_Quaternion_Orders_over_Polynomial_Rings (H89E7)

      Creation of Quaternion Orders over Number Rings
            TameOrder(A) : AlgQuat[FldAlg] -> AlgAssVOrd
            MaximalOrder(O) : AlgAssVOrd[RngOrd] -> AlgAssVOrd
            pMaximalOrder(O, p) : AlgAssVOrd, RngOrdIdl -> AlgAssVOrd, RngIntElt
            IsMaximal(O) : AlgAssVOrd -> BoolElt
            IspMaximal(O, p) : AlgAssVOrd, RngOrdIdl -> BoolElt
            IsHereditary(O) : AlgAssVOrd -> BoolElt

 
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 (H89E8)

 
Attributes of Quaternion Algebras
      BaseField(A) : AlgQuat -> Fld
      Basis(A) : AlgQuat -> SeqEnum
      Discriminant(A) : AlgQuat[FldRat] -> RngElt
      RamifiedPrimes(A) : AlgQuat -> SeqEnum
      Example AlgQuat_Ramified_Primes (H89E9)
      RamifiedPlaces(A) : AlgQuat[FldAlg] -> SeqEnum, SeqEnum
      StandardForm(A) : AlgQuat -> RngElt, RngElt, AlgQuat, Map

 
Hilbert Symbols and Embeddings
      HilbertSymbol(a, b, p) : FldRatElt, FldRatElt, RngIntElt -> RngIntElt
      IsRamified(p, A) : RngIntElt, AlgQuat[FldRat] -> BoolElt
      Example AlgQuat_Hilbert_Symbols (H89E10)
      pMatrixRing(A, p) : AlgQuat, RngOrdIdl -> AlgQuat, Map, Map
      IsSplittingField(K, A) : FldAlg, AlgQuat -> BoolElt, .
      Embed(K, A) : FldAlg, AlgQuat -> AlgQuatElt, Map
      Embed(Oc, O) : RngOrd, AlgAssVOrd -> AlgAssVOrdElt, Map
      Example AlgQuat_Embed (H89E11)

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

 
Recognition Functions
      IsMatrixRing(A) : AlgQuat[FldAlg] -> BoolElt
      MatrixRing(A, eps) : AlgQuat, AlgQuatElt -> AlgQuat, Map
      Example AlgQuat_Quaternion_MatrixRing (H89E12)
      IsQuaternionAlgebra(A) : AlgAss -> BoolElt, AlgQuat, Map
      Example AlgQuat_Quaternion_IsQuaternionAlgebra (H89E13)
      MatrixRepresentation(A) : AlgQuat -> Map

 
Attributes of Orders
      QuaternionAlgebra(S) : AlgQuatOrd -> AlgQuat
      BasisMatrix(A) : AlgQuatOrd -> AlgMatElt
      EmbeddingMatrix(S) : AlgQuatOrd -> AlgMatElt
      Discriminant(S) : AlgQuatOrd -> RngElt
      FactoredDiscriminant(S) : AlgQuatOrd -> SeqEnum
      Conductor(A) : AlgQuatOrd -> RngIntElt
      Level(S) : AlgQuatOrd -> RngIntElt

 
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, [AlgQuatOrdElt] -> AlgQuatOrdIdl
            PrimeIdeal(S, p) : AlgQuatOrd, RngIntElt -> AlgQuatOrdIdl
            CommutatorIdeal(S) : AlgQuatOrd -> AlgQuatOrdIdl
            Example AlgQuat_Elementary_Ideals (H89E14)
            Example AlgQuat_Ideal_Bases (H89E15)
            LeftOrder(I) : AlgQuatOrdIdl -> AlgQuatOrd
            RightOrder(I) : AlgQuatOrdIdl -> AlgQuatOrd
            Example AlgQuat_Left_Right_Quaternion_Ordre (H89E16)

      Enumeration of Ideal Classes
            LeftIdealClasses(S) : AlgQuatOrd -> [AlgQuatOrdIdl]
            TwoSidedIdealClasses(S) : AlgQuatOrd -> [AlgQuatOrdIdl]
            TwoSidedIdealClassGroup(S : Support) : AlgAssVOrd[RngOrd] -> GrpAb, Map
            ConjugacyClasses(S) : AlgAssVOrd -> SeqEnum
            Example AlgQuat_Ideal_Enumeration (H89E17)
            Example AlgQuat_Ideal_Enumeration (H89E18)
            Example AlgQuat_Ideal_Enumeration (H89E19)

      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
      GramMatrix(S) : AlgQuatOrd -> AlgMatElt
      ReducedBasis(S) : AlgQuatOrd -> SeqEnum
      ReducedGramMatrix(S) : AlgQuatOrd -> AlgMatElt
      Example AlgQuat_Basis_Reduction (H89E20)
      OptimizedRepresentation(O) : AlgAssVOrd -> AlgQuat, Map
      OptimizedRepresentation(A) : AlgQuat -> AlgQuat, Map
      Enumerate(O, A, B) : AlgAssVOrd[RngOrd], RngElt, RngElt -> [AlgAssVOrdElt]
      ReducedBasis(O) : AlgAssVOrd[RngOrd] -> [AlgAssVElt]

 
Isomorphisms
      IsIsomorphic(A, B) : AlgQuat[FldRat], AlgQuat[FldRat] -> BoolElt
      IsIsomorphic(S, T) : AlgQuatOrd, AlgQuatOrd -> BoolElt, Map, AlgQuatElt
      IsIsomorphic(I, J) : AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> BoolElt, AlgAssVElt
      Isomorphism(S, T) : AlgQuatOrd, AlgQuatOrd -> Map
      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
      IsPrincipal(I) : AlgAssVOrdIdl[RngOrd] -> BoolElt, AlgQuatElt
      Example AlgQuat_Isomorphism_example (H89E21)
      Example AlgQuat_Left_Right_Isomorphisms (H89E22)
      Example AlgQuat_Left_Right_Isomorphisms_Number_Field (H89E23)

 
Units and Unit Groups
      NormOneGroup(S) : AlgAssVOrd -> GrpPerm, Map
      Units(S) : AlgQuatOrd -> SeqEnum
      MultiplicativeGroup(S) : AlgQuatOrd -> GrpPerm, Map
      Example AlgQuat_Unit_Group (H89E24)

 
Bibliography

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