QUATERNION ALGEBRAS

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QUATERNION ALGEBRAS

Acknowledgements

Introduction

Creation of Quaternion Algebras

Creation of Quaternion Orders

Elements of Quaternion Algebras
Creation of Elements
Arithmetic of Elements

Attributes of Quaternion Algebras

Predicates on Algebras

Attributes of Orders and Ideals

Ideal Theory of Orders
Creation and Access Functions
Enumeration of Ideal Classes
Operations on Ideals

Norm Spaces and Basis Reduction

Order and Ideal Isomorphisms

Units and Unit Groups

Bibliography

DETAILS

Introduction

Creation of Quaternion Algebras
QuaternionAlgebra< K | a, b > : Rng, RngElt, RngElt -> AlgQuat
AssignNames(~A,S) : AlgQuat, [MonStgElt] ->
Example AlgQuat_Quaternion_Constructor (H68E1)
QuaternionAlgebra(N) : RngIntElt -> AlgQuat
QuaternionAlgebra(D1, D2, T) : RngIntElt, RngIntElt, RngIntElt -> AlgQuat
Example AlgQuat_Quaternion_Constructor_over_Rationals (H68E2)

Creation of Quaternion Orders
MaximalOrder(A) : AlgQuat -> AlgQuatOrd
QuaternionOrder(A,M) : AlgQuat, RngIntElt -> AlgQuatOrd
QuaternionOrder(N) : RngIntElt -> AlgQuatOrd
QuaternionOrder(D1, D2, T) : RngIntElt, RngIntElt, RngIntElt -> AlgQuat
Example AlgQuat_Quaternion_Orders_over_the_Integers (H68E3)
QuaternionOrder(S) : [AlgQuatElt] -> AlgQuatOrd
QuaternionOrder(R,S) : Rng, [AlgQuatElt] -> AlgQuatOrd
Example AlgQuat_Quaternion_Orders_over_Polynomial_Rings (H68E4)

Elements of Quaternion Algebras

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

Attributes of Quaternion Algebras
BaseField(A) : AlgQuat -> Fld
Basis(A) : AlgQuat -> SeqEnum
Discriminant(A) : AlgQuat -> FldRatElt
RamifiedPrimes(A) : AlgQuat -> SeqEnum
Example AlgQuat_Ramified_Primes (H68E6)

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

Attributes of Orders and Ideals
QuaternionAlgebra(S) : AlgQuatOrd -> AlgQuat
BaseRing(S) : AlgQuatOrd -> Rng
Basis(S) : AlgQuatOrd -> SeqEnum
EmbeddingMatrix(S) : AlgQuatOrd -> AlgMatElt
Discriminant(S) : AlgQuatOrd -> RngIntElt
Level(S) : AlgQuatOrd -> RngIntElt
Example AlgQuat_Ideal_Bases (H68E7)

Ideal Theory of Orders

Creation and Access Functions
LeftOrder(I) : AlgQuatOrd -> AlgQuatOrd
RightOrder(I) : AlgQuatOrd -> AlgQuatOrd
LeftIdeal(S, X) : AlgQuatOrd, [AlgQuatOrdElt] -> AlgQuatOrd
RightIdeal(S, X) : AlgQuatOrd, [AlgQuatOrdElt] -> AlgQuatOrd
ideal<S | X> : AlgQuatOrd, [AlgQuatOrdElt] -> AlgQuatOrd
PrimeIdeal(S, p) : AlgQuatOrd, RngIntElt -> AlgQuatOrd
CommutatorIdeal(S) : AlgQuatOrd -> AlgQuatOrd
Example AlgQuat_Elementary_Ideals (H68E8)

Enumeration of Ideal Classes
LeftIdealClasses(S) : AlgQuatOrd -> [AlgQuatOrd]
RightIdealClasses(S) : AlgQuatOrd -> [AlgQuatOrd]
TwoSidedIdealClasses(S) : AlgQuatOrd -> [AlgQuatOrd]
Example AlgQuat_Ideal_Enumeration (H68E9)

Operations on Ideals
I * J : AlgQuatOrd, AlgQuatOrd -> AlgQuatOrd
Conjugate(I) : AlgQuatOrd -> AlgQuatOrd
I meet J : AlgQuatOrd, AlgQuatOrd -> AlgQuatOrd
Norm(I) : AlgQuatOrd -> RngIntElt
Example AlgQuat_Ideal_Arithmetic (H68E10)

Norm Spaces and Basis Reduction
NormSpace(A) : AlgQuat -> ModTupFld
GramMatrix(S) : AlgQuatOrd -> AlgMat
ReducedBasis(S) : AlgQuatOrd -> SeqEnum
ReducedGramMatrix(S) : AlgQuatOrd -> AlgMat
Example AlgQuat_Basis_Reduction (H68E11)

Order and Ideal Isomorphisms
IsIsomorphic(I,J) : AlgQuatOrd, AlgQuatOrd -> BoolElt
Isomorphism(I,J) : AlgQuatOrd, AlgQuatOrd -> BoolElt
IsLeftIsomorphic(I,J) : AlgQuatOrd, AlgQuatOrd -> BoolElt, Map, AlgQuatElt
LeftIsomorphism(I,J) : AlgQuatOrd, AlgQuatOrd -> Map, AlgQuatElt
IsRightIsomorphic(I,J) : AlgQuatOrd, AlgQuatOrd -> BoolElt, Map, AlgQuatElt
RightIsomorphism(I,J) : AlgQuatOrd, AlgQuatOrd -> Map, AlgQuatElt
Example AlgQuat_Left_Right_Isomorphisms (H68E12)

Units and Unit Groups
Units(S) : AlgQuatOrd -> SeqEnum
UnitGroup(S) : AlgQuatOrd -> GrpPerm, Map

Bibliography