Quaternion Algebras
- Introduction
- Creation of Quaternion Algebras
QuaternionAlgebra< K | a, b >: Rng, RngElt, RngElt → AlgQuatAssignNames(~A, S): AlgQuat, [MonStgElt]Example: Quaternion ConstructorExample: Quaternion Constructor Char2QuaternionAlgebra(N): RngIntElt → AlgQuatQuaternionAlgebra(N): RngUPolElt → AlgQuatQuaternionAlgebra(I): RngOrdIdl → AlgQuatQuaternionAlgebra(I, S): RngOrdIdl, [PlcNumElt] → AlgQuatQuaternionAlgebra(S): [PlcNumElt] → AlgQuatExample: Quaternion Constructor Over NumberFieldQuaternionAlgebra(D1, D2, T): RngIntElt, RngIntElt, RngIntElt → AlgQuatExample: Quaternion Constructor Over Rationals
- Creation of Quaternion Orders
- Elements of Quaternion Algebras
- Attributes of Quaternion Algebras
- Hilbert Symbols and Embeddings
HilbertSymbol(a, b, p): FldRatElt, FldRatElt, RngIntElt → RngIntEltHilbertSymbol(a, b, p): FldFunRatElt, FldFunRatElt, RngElt p → RngIntEltHilbertSymbol(a, b, p): FldNumElt, FldNumElt, RngOrdIdl → RngIntEltHilbertSymbol(A, p): AlgQuat[FldRat], RngIntElt → RngIntEltHilbertSymbol(A, p): AlgQuat[FldFunRat], RngElt → RngIntEltHilbertSymbol(A, p): AlgQuat, RngOrdIdl → RngIntEltIsRamified(p, A): RngElt, AlgQuat → BoolEltIsUnramified(p, A): RngElt, AlgQuat → BoolEltIsRamified(p, A): RngUPol, AlgQuat[FldFunRat] → BoolEltIsUnramified(p, A): RngUPol, AlgQuat[FldFunRat] → BoolEltIsRamified(p, A): RngOrdIdl, AlgQuat[FldAlg] → BoolEltIsUnramified(p, A): RngOrdIdl, AlgQuat[FldAlg] → BoolEltExample: Hilbert SymbolspMatrixRing(A, p): AlgQuat, RngOrdIdl → AlgMat, Map, MappMatrixRing(A, p): AlgQuat, RngElt → AlgMat, Map, MappMatrixRing(O, p): AlgAssVOrd, RngOrdIdl → AlgMat, Map, MappMatrixRing(O, p): AlgQuatOrd, RngElt → AlgMat, Map, MappMatrixRing(O, p): AlgQuatOrd[RngInt], RngInt → AlgMat, Map, MapIsSplittingField(K, A): Fld, AlgQuat → BoolElt, AlgQuatElt, MapHasEmbedding(K, A): Fld, AlgQuat → BoolElt, AlgQuatElt, MapEmbed(K, A): Fld, AlgQuat → AlgQuatElt, MapEmbed(Oc, O): RngOrd, AlgAssVOrd → AlgAssVOrdElt, MapExample: Embed
- Predicates on Algebras
- Recognition Functions
IsMatrixRing(A): AlgQuat → BoolElt, AlgMat, MapMatrixRing(A, eps): AlgQuat, AlgQuatElt → AlgMat, MapMatrixAlgebra(A, eps): AlgQuat, AlgQuatElt → AlgMat, MapExample: Quaternion MatrixRingIsQuaternionAlgebra(B): AlgAss → BoolElt, AlgQuat, MapIsQuaternionAlgebra(B): AlgMat → BoolElt, AlgQuat, MapExample: Quaternion IsQuaternionAlgebraMatrixRepresentation(A): AlgQuat → MapMatrixRepresentation(R): AlgQuatOrd → Map
- Attributes of Orders
- Predicates of Orders
IsMaximal(O): AlgAssVOrd → BoolEltIspMaximal(O, p): AlgAssVOrd, RngOrdIdl → BoolEltIspMaximal(O, p): AlgQuatOrd, RngElt → BoolEltIsEichler(O): AlgAssVOrd → BoolElt, AlgAssVOrd, AlgAssVOrdIsEichler(O, p): AlgAssVOrd, RngOrdIdl → BoolElt, AlgAssVOrd, AlgAssVOrdIsEichler(O, p): AlgQuatOrd, RngElt → BoolElt, AlgQuatOrd, AlgQuatOrdEichlerInvariant(O, p): AlgAssVOrd, RngOrdIdl → RngIntEltEichlerInvariant(O, p): AlgQuatOrd, RngElt → RngIntEltIsHereditary(O): AlgAssVOrd → BoolEltIsHereditary(O, p): AlgAssVOrd, RngOrdIdl → BoolEltIsHereditary(O, p): AlgQuatOrd, RngElt → BoolEltIsGorenstein(O): AlgAssVOrd → BoolElt, .IsGorenstein(O, p): AlgAssVOrd, RngOrdIdl → BoolElt, RngIntEltIsGorenstein(O, p): AlgQuatOrd, RngElt → BoolElt, RngIntEltIsBass(O): AlgAssVOrd → BoolEltIsBass(O, p): AlgAssVOrd, RngOrdIdl → BoolEltIsBass(O, p): AlgQuatOrd, RngElt → BoolEltIsSameType(O1, O2): AlgAssVOrd, AlgAssVOrd → BoolElt
- Operations with Orders
- Ideal Theory of Orders
- Creation and Access Functions
LeftIdeal(S, X): AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdllideal<S | X>: AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdlRightIdeal(S, X): AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdlrideal<S | X>: AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdideal<S | X>: AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdlPrimeIdeal(S, p): AlgQuatOrd, RngElt → AlgQuatOrdIdlCommutatorIdeal(S): AlgQuatOrd → AlgQuatOrdIdlCommutatorIdeal(S): AlgAssVOrd → AlgAssVOrdIdlMaximalLeftIdeals(O, p): AlgQuatOrd, RngElt → [AlgQuatOrdIdl]MaximalLeftIdeals(O, p): AlgAssVOrd, RngOrdIdl → [AlgAssVOrdIdl]MaximalRightIdeals(O, p): AlgQuatOrd, RngElt → [AlgQuatOrdIdl]MaximalRightIdeals(O, p): AlgAssVOrd, RngOrdIdl → [AlgAssVOrdIdl]Example: Elementary IdealsExample: Ideal BasesLeftOrder(I): AlgQuatOrdIdl → AlgQuatOrdLeftOrder(I): AlgAssVOrdIdl[RngOrd] → AlgAssVOrdRightOrder(I): AlgQuatOrdIdl → AlgQuatOrdRightOrder(I): AlgAssVOrdIdl[RngOrd] → AlgAssVOrdExample: Left Right Quaternion Ordre
- Enumeration of Ideal Classes
- Operations on Ideals
- Norm Spaces and Basis Reduction
NormSpace(A): AlgQuat → ModTupFld, MapNormSpace(S): AlgQuatOrd → ModTupRng, MapNormModule(S): AlgQuatOrd → ModTupRng, MapGramMatrix(S): AlgQuatOrd → AlgMatEltGramMatrix(I): AlgQuatOrdIdl → AlgMatEltReducedGramMatrix(S): AlgQuatOrd[RngInt] → AlgMatEltReducedGramMatrix(S): AlgQuatOrdIdl[RngInt] → AlgMatEltReducedBasis(S): AlgQuatOrd[RngInt] → SeqEnumReducedBasis(S): AlgQuatOrdIdl[RngInt] → SeqEnumExample: Basis ReductionReducedGramMatrix(S): AlgQuatOrd[RngUPol] → AlgMatElt, SeqEnumReducedGramMatrix(S): AlgQuatOrdIdl[RngUPol] → AlgMatElt, SeqEnumReducedBasis(S): AlgQuatOrd[RngUPol] → SeqEnumReducedBasis(S): AlgQuatOrdIdl[RngUPol] → SeqEnumReducedBasis(O): AlgAssVOrd[RngOrd] → [AlgAssVElt]ReducedBasis(I): AlgAssVOrdIdl[RngOrd] → [AlgAssVOrdElt]OptimizedRepresentation(O): AlgAssVOrd → AlgQuat, MapOptimisedRepresentation(O): AlgAssVOrd → AlgQuat, MapOptimizedRepresentation(A): AlgQuat → AlgQuat, MapOptimisedRepresentation(A): AlgQuat → AlgQuat, MapEnumerate(O, A, B): AlgQuatOrd[RngInt], RngIntElt, RngIntElt → [AlgQuatOrdElt]Enumerate(O, B): AlgQuatOrd[RngInt], RngIntElt → [AlgQuatOrdElt]Enumerate(O, A, B): AlgQuatOrdIdl[RngInt], RngElt, RngElt → [AlgQuatElt]Enumerate(O, B): AlgQuatOrd[RngInt], RngElt → [AlgQuatElt]Enumerate(O, A, B): AlgAssVOrd[RngOrd], RngElt, RngElt → [AlgAssVOrdElt]Enumerate(O, B): AlgAssVOrd[RngOrd], RngElt → [AlgAssVOrdElt]Enumerate(O, B): AlgAssVOrd[RngOrd], [RngElt] → [AlgAssVOrdElt]Enumerate(I, B): AlgAssVOrdIdl[RngOrd], [RngElt] → [AlgAssVOrdElt]
- Isomorphisms
- Isomorphisms of Algebras
- Isomorphisms of Orders
IsIsomorphic(S, T): AlgQuatOrd, AlgQuatOrd → BoolElt, Map, AlgQuatEltIsIsomorphic(S, T): AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] → BoolElt, Map, AlgQuatEltIsConjugate(S, T): AlgAssVOrd, AlgAssVOrd → BoolElt, Map, AlgQuatEltIsomorphism(S, T): AlgQuatOrd, AlgQuatOrd → Map
- Isomorphisms of Ideals
IsIsomorphic(I, J): AlgAssVOrdIdl, AlgAssVOrdIdl → BoolElt, AlgAssVEltIsPrincipal(I): AlgAssVOrdIdl → BoolElt, AlgQuatEltIsLeftIsomorphic(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → BoolElt, Map, AlgQuatEltIsRightIsomorphic(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → BoolElt, Map, AlgQuatEltIsLeftIsomorphic(I, J): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] → BoolElt, AlgQuatEltIsRightIsomorphic(I, J): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] → BoolElt, AlgQuatEltLeftIsomorphism(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → Map, AlgQuatEltRightIsomorphism(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → Map, AlgQuatElt
- Examples
- Units and Unit Groups
