Creation of Quaternion Algebras
QuaternionAlgebra< K | a, b > : Rng, RngElt, RngElt -> AlgQuat
AssignNames(~A, S) : AlgQuat, [MonStgElt] ->
Example AlgQuat_Quaternion_Constructor (H86E1)
Example AlgQuat_Quaternion_Constructor_char2 (H86E2)
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 (H86E3)
QuaternionAlgebra(D1, D2, T) : RngIntElt, RngIntElt, RngIntElt -> AlgQuat
Example AlgQuat_Quaternion_Constructor_over_Rationals (H86E4)
Creation of Orders from Elements
QuaternionOrder(S) : [AlgQuatElt] -> AlgQuatOrd
QuaternionOrder(R, S) : Rng, [AlgQuatElt] -> AlgQuatOrd
Example AlgQuat_Quaternion_Orders_over_Q_FqX (H86E5)
Creation of Maximal Orders
MaximalOrder(A) : AlgQuat[FldRat] -> AlgQuatOrd
Example AlgQuat_Quaternion_MaximalOrder (H86E6)
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 (H86E7)
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 (H86E8)
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 (H86E9)
Attributes of Quaternion Algebras
BaseField(A) : AlgQuat -> Fld
Basis(A) : AlgQuat -> SeqEnum
RamifiedPrimes(A) : AlgQuat -> SeqEnum
Example AlgQuat_Ramified_Primes (H86E10)
RamifiedPlaces(A) : AlgQuat -> SeqEnum, SeqEnum
Example AlgQuat_Ramified_Primes_FqX (H86E11)
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 (H86E12)
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 (H86E13)
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 (H86E14)
IsQuaternionAlgebra(B) : AlgAss -> BoolElt, AlgQuat, Map
Example AlgQuat_Quaternion_IsQuaternionAlgebra (H86E15)
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
Operations with Orders
O1 meet O2 : AlgQuatOrd[RngInt], AlgQuatOrd[RngInt] -> AlgQuatOrd
O ^ x : AlgQuatOrd, AlgQuatElt -> AlgQuatOrd
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 (H86E16)
Example AlgQuat_Ideal_Bases (H86E17)
LeftOrder(I) : AlgQuatOrdIdl -> AlgQuatOrd
RightOrder(I) : AlgQuatOrdIdl -> AlgQuatOrd
Example AlgQuat_Left_Right_Quaternion_Ordre (H86E18)
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 (H86E19)
Example AlgQuat_Ideal_Enumeration (H86E20)
Example AlgQuat_Ideal_Enumeration (H86E21)
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 (H86E22)
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 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 (H86E23)
Example AlgQuat_Isomorphism_example (H86E24)
Example AlgQuat_Left_Right_Isomorphisms (H86E25)
Example AlgQuat_Left_Right_Isomorphisms_Number_Field (H86E26)
Units and Unit Groups
NormOneGroup(S) : AlgAssVOrd -> GrpPerm, Map
Units(S) : AlgQuatOrd -> SeqEnum
MultiplicativeGroup(S) : AlgQuatOrd[RngInt] -> GrpPerm, Map
Example AlgQuat_Unit_Group (H86E27)
Example AlgQuat_Unit_Group_NumberRing (H86E28)
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013