# Quaternion Algebras - [Introduction](introduction.md) - [Creation of Quaternion Algebras](creation.md) - [`QuaternionAlgebra< K | a, b >: Rng, RngElt, RngElt → AlgQuat`](creation.md#constructor-constructor-quaternionalgebra-rng-rngelt-rngelt-algquat) - [`AssignNames(~A, S): AlgQuat, [MonStgElt]`](creation.md#function-assignnames-algquat-monstgelt-ref) - [`Example: Quaternion Constructor`](creation.md#example-ex-24b190) - [`Example: Quaternion Constructor Char2`](creation.md#example-ex-c9a2d5) - [`QuaternionAlgebra(N): RngIntElt → AlgQuat`](creation.md#function-quaternionalgebra-rngintelt) - [`QuaternionAlgebra(N): RngUPolElt → AlgQuat`](creation.md#function-quaternionalgebra-rngupolelt) - [`QuaternionAlgebra(I): RngOrdIdl → AlgQuat`](creation.md#function-quaternionalgebra-rngordidl) - [`QuaternionAlgebra(I, S): RngOrdIdl, [PlcNumElt] → AlgQuat`](creation.md#function-quaternionalgebra-rngordidl-plcnumelt) - [`QuaternionAlgebra(S): [PlcNumElt] → AlgQuat`](creation.md#function-quaternionalgebra-plcnumelt) - [`Example: Quaternion Constructor Over NumberField`](creation.md#example-ex-b19daa) - [`QuaternionAlgebra(D1, D2, T): RngIntElt, RngIntElt, RngIntElt → AlgQuat`](creation.md#function-quaternionalgebra-rngintelt-rngintelt-rngintelt) - [`Example: Quaternion Constructor Over Rationals`](creation.md#example-ex-359d98) - [Creation of Quaternion Orders](order-creation.md) - [Creation of Orders from Elements](order-creation.md#creation-of-orders-from-elements) - [`QuaternionOrder(S): [AlgQuatElt] → AlgQuatOrd`](order-creation.md#function-quaternionorder-algquatelt) - [`QuaternionOrder(R, S): Rng, [AlgQuatElt] → AlgQuatOrd`](order-creation.md#function-quaternionorder-rng-algquatelt) - [`Order(R, S): Rng, [AlgQuatElt] → AlgQuatOrd`](order-creation.md#function-order-rng-algquatelt) - [`Example: Quaternion Orders Over Q FqX`](order-creation.md#example-ex-b0f32f) - [Creation of Maximal Orders](order-creation.md#creation-of-maximal-orders) - [`MaximalOrder(A): AlgQuat[FldRat] → AlgQuatOrd`](order-creation.md#function-maximalorder-algquat-fldrat) - [`MaximalOrder(A): AlgQuat[FldFunRat] → AlgQuatOrd`](order-creation.md#function-maximalorder-algquat-fldfunrat) - [`MaximalOrder(A): AlgQuat[FldOrd] → AlgAssVOrd`](order-creation.md#function-maximalorder-algquat-fldord) - [`MaximalOrder(A): AlgQuat[FldNum] → AlgAssVOrd`](order-creation.md#function-maximalorder-algquat-fldnum) - [`Example: Quaternion MaximalOrder`](order-creation.md#example-ex-9b04e9) - [`MaximalOrder(O): AlgQuatOrd → AlgQuat`](order-creation.md#function-maximalorder-algquatord) - [`MaximalOrder(O): AlgAssVOrd[RngOrd] → AlgAssVOrd`](order-creation.md#function-maximalorder-algassvord-rngord) - [`pMaximalOrder(O, p): AlgQuatOrd, RngElt → AlgQuatOrd, RngIntElt`](order-creation.md#function-pmaximalorder-algquatord-rngelt) - [`pMaximalOrder(O, p): AlgAssVOrd, RngOrdIdl → AlgAssVOrd, RngIntElt`](order-creation.md#function-pmaximalorder-algassvord-rngordidl) - [`TameOrder(A): AlgQuat[FldAlg] → AlgAssVOrd`](order-creation.md#function-tameorder-algquat-fldalg) - [Creation of Orders with given Discriminant](order-creation.md#creation-of-orders-with-given-discriminant) - [`Order(O, N): AlgQuatOrd, RngElt → AlgQuatOrd`](order-creation.md#function-order-algquatord-rngelt) - [`Order(O, N): AlgAssVOrd, RngOrdIdl → AlgAssVOrd`](order-creation.md#function-order-algassvord-rngordidl) - [`GorensteinClosure(O): AlgAssVOrd → AlgAssVOrd, .`](order-creation.md#function-algquat-gorensteinclosure) - [`Example: Quaternion Orders`](order-creation.md#example-ex-966516) - [Creation of Orders with given Discriminant over the Integers](order-creation.md#creation-of-orders-with-given-discriminant-over-the-integers) - [`QuaternionOrder(A, M): AlgQuat[FldRat], RngIntElt → AlgQuatOrd`](order-creation.md#function-quaternionorder-algquat-fldrat-rngintelt) - [`QuaternionOrder(N): RngIntElt → AlgQuatOrd`](order-creation.md#function-quaternionorder-rngintelt) - [`QuaternionOrder(N, M): RngIntElt, RngIntElt → AlgQuatOrd`](order-creation.md#function-quaternionorder-rngintelt-rngintelt) - [`QuaternionOrder(D1, D2, T): RngIntElt, RngIntElt, RngIntElt → AlgQuat`](order-creation.md#function-quaternionorder-rngintelt-rngintelt-rngintelt) - [`Example: Quaternion Orders Over The Integers`](order-creation.md#example-ex-f04e9e) - [Elements of Quaternion Algebras](elements.md) - [Creation of Elements](elements.md#creation-of-elements) - [`A ! 0: AlgQuat, RngIntElt → AlgQuatElt`](elements.md#operation-op-algquat-rngintelt) - [`Zero(A): AlgQuat → AlgQuatElt`](elements.md#function-zero-algquat) - [`A ! 1: AlgQuat, RngIntElt → AlgQuatElt`](elements.md#operation-op-algquat-rngintelt-2) - [`One(A): AlgQuat → AlgQuatElt`](elements.md#function-one-algquat) - [`A . i: AlgQuat, RngIntElt → AlgQuatElt`](elements.md#operation-operation-algquat-rngintelt-algquatelt) - [`Name(A, i): AlgQuat, RngIntElt → AlgQuatElt`](elements.md#function-name-algquat-rngintelt) - [`A ! x: AlgQuat, Any → AlgQuatElt`](elements.md#operation-op-algquat-any) - [Arithmetic of Elements](elements.md#arithmetic-of-elements) - [`x + y: AlgQuatElt, AlgQuatElt → AlgQuatElt`](elements.md#operation-op-plus-algquatelt-algquatelt) - [`x - y: AlgQuatElt, AlgQuatElt → AlgQuatElt`](elements.md#operation-op-minus-algquatelt-algquatelt) - [`x * y: AlgQuatElt, AlgQuatElt → AlgQuatElt`](elements.md#operation-op-times-algquatelt-algquatelt) - [`x / y: AlgQuatElt, AlgQuatElt → AlgQuatElt`](elements.md#operation-op-div-algquatelt-algquatelt) - [`x / y: AlgQuatOrdElt, AlgQuatOrdElt → AlgQuatElt`](elements.md#operation-op-div-algquatordelt-algquatordelt) - [`x eq y: AlgQuatElt, AlgQuatElt → BoolElt`](elements.md#operation-op-eq-algquatelt-algquatelt) - [`x ne y: AlgQuatElt, AlgQuatElt → BoolElt`](elements.md#operation-op-ne-algquatelt-algquatelt) - [`x in A: AlgQuatElt, AlgQuat → BoolElt`](elements.md#operation-op-in-algquatelt-algquat) - [`x notin A: AlgQuatElt, AlgQuat → BoolElt`](elements.md#operation-op-notin-algquatelt-algquat) - [`Conjugate(x): AlgQuatElt → AlgQuatElt`](elements.md#function-conjugate-algquatelt) - [`Conjugate(x): AlgQuatOrdElt → AlgQuatOrdElt`](elements.md#function-conjugate-algquatordelt) - [`Conjugate(x): AlgAssVOrdElt → AlgAssVOrdElt`](elements.md#function-conjugate-algassvordelt) - [`ElementToSequence(x): AlgQuatElt → SeqEnum`](elements.md#function-elementtosequence-algquatelt) - [`Eltseq(x): AlgQuatElt → SeqEnum`](elements.md#function-eltseq-algquatelt) - [`Coordinates(x): AlgQuatElt → SeqEnum`](elements.md#function-coordinates-algquatelt) - [`Norm(x): AlgQuatElt → FldElt`](elements.md#function-norm-algquatelt) - [`Norm(x): AlgQuatOrdElt → RngElt`](elements.md#function-norm-algquatordelt) - [`Trace(x): AlgQuatElt → FldElt`](elements.md#function-trace-algquatelt) - [`Trace(x): AlgQuatOrdElt → RngElt`](elements.md#function-trace-algquatordelt) - [`CharacteristicPolynomial(x): AlgQuatElt → RngUPolElt`](elements.md#function-characteristicpolynomial-algquatelt) - [`CharacteristicPolynomial(x): AlgQuatOrdElt → RngUPolElt`](elements.md#function-characteristicpolynomial-algquatordelt) - [`MinimalPolynomial(x): AlgQuatElt → RngUPolElt`](elements.md#function-minimalpolynomial-algquatelt) - [`Example: Element Arithmetic`](elements.md#example-ex-317c57) - [Attributes of Quaternion Algebras](algebra-attributes.md) - [`BaseField(A): AlgQuat → Fld`](algebra-attributes.md#function-basefield-algquat) - [`BaseRing(A): AlgQuat → Fld`](algebra-attributes.md#function-basering-algquat) - [`Basis(A): AlgQuat → SeqEnum`](algebra-attributes.md#function-basis-algquat) - [`RamifiedPrimes(A): AlgQuat → SeqEnum`](algebra-attributes.md#function-ramifiedprimes-algquat) - [`Example: Ramified Primes`](algebra-attributes.md#example-ex-4e1f00) - [`RamifiedPlaces(A): AlgQuat → SeqEnum, SeqEnum`](algebra-attributes.md#function-ramifiedplaces-algquat) - [`FactoredDiscriminant(A): AlgQuat → SeqEnum, SeqEnum`](algebra-attributes.md#function-factoreddiscriminant-algquat) - [`Example: Ramified Primes FqX`](algebra-attributes.md#example-ex-642861) - [`Discriminant(A): AlgQuat[FldRat] → RngIntElt`](algebra-attributes.md#function-discriminant-algquat-fldrat) - [`Discriminant(A): AlgQuat[FldFunRat] → RngUPolElt`](algebra-attributes.md#function-discriminant-algquat-fldfunrat) - [`Discriminant(A): AlgQuat[FldAlg] → RngOrdIdl, SeqEnum`](algebra-attributes.md#function-discriminant-algquat-fldalg) - [`StandardForm(A): AlgQuat → RngElt, RngElt, AlgQuat, Map`](algebra-attributes.md#function-standardform-algquat) - [Hilbert Symbols and Embeddings](hilbert-symbols.md) - [`HilbertSymbol(a, b, p): FldRatElt, FldRatElt, RngIntElt → RngIntElt`](hilbert-symbols.md#function-hilbertsymbol-fldratelt-fldratelt-rngintelt) - [`HilbertSymbol(a, b, p): FldFunRatElt, FldFunRatElt, RngElt p → RngIntElt`](hilbert-symbols.md#function-hilbertsymbol-fldfunratelt-fldfunratelt-rngelt-p) - [`HilbertSymbol(a, b, p): FldNumElt, FldNumElt, RngOrdIdl → RngIntElt`](hilbert-symbols.md#function-hilbertsymbol-fldnumelt-fldnumelt-rngordidl) - [`HilbertSymbol(A, p): AlgQuat[FldRat], RngIntElt → RngIntElt`](hilbert-symbols.md#function-hilbertsymbol-algquat-fldrat-rngintelt) - [`HilbertSymbol(A, p): AlgQuat[FldFunRat], RngElt → RngIntElt`](hilbert-symbols.md#function-hilbertsymbol-algquat-fldfunrat-rngelt) - [`HilbertSymbol(A, p): AlgQuat, RngOrdIdl → RngIntElt`](hilbert-symbols.md#function-hilbertsymbol-algquat-rngordidl) - [`IsRamified(p, A): RngElt, AlgQuat → BoolElt`](hilbert-symbols.md#function-isramified-rngelt-algquat) - [`IsUnramified(p, A): RngElt, AlgQuat → BoolElt`](hilbert-symbols.md#function-isunramified-rngelt-algquat) - [`IsRamified(p, A): RngUPol, AlgQuat[FldFunRat] → BoolElt`](hilbert-symbols.md#function-isramified-rngupol-algquat-fldfunrat) - [`IsUnramified(p, A): RngUPol, AlgQuat[FldFunRat] → BoolElt`](hilbert-symbols.md#function-isunramified-rngupol-algquat-fldfunrat) - [`IsRamified(p, A): RngOrdIdl, AlgQuat[FldAlg] → BoolElt`](hilbert-symbols.md#function-isramified-rngordidl-algquat-fldalg) - [`IsUnramified(p, A): RngOrdIdl, AlgQuat[FldAlg] → BoolElt`](hilbert-symbols.md#function-isunramified-rngordidl-algquat-fldalg) - [`Example: Hilbert Symbols`](hilbert-symbols.md#example-ex-9d678c) - [`pMatrixRing(A, p): AlgQuat, RngOrdIdl → AlgMat, Map, Map`](hilbert-symbols.md#function-pmatrixring-algquat-rngordidl) - [`pMatrixRing(A, p): AlgQuat, RngElt → AlgMat, Map, Map`](hilbert-symbols.md#function-pmatrixring-algquat-rngelt) - [`pMatrixRing(O, p): AlgAssVOrd, RngOrdIdl → AlgMat, Map, Map`](hilbert-symbols.md#function-pmatrixring-algassvord-rngordidl) - [`pMatrixRing(O, p): AlgQuatOrd, RngElt → AlgMat, Map, Map`](hilbert-symbols.md#function-pmatrixring-algquatord-rngelt) - [`pMatrixRing(O, p): AlgQuatOrd[RngInt], RngInt → AlgMat, Map, Map`](hilbert-symbols.md#function-pmatrixring-algquatord-rngint-rngint) - [`IsSplittingField(K, A): Fld, AlgQuat → BoolElt, AlgQuatElt, Map`](hilbert-symbols.md#function-issplittingfield-fld-algquat) - [`HasEmbedding(K, A): Fld, AlgQuat → BoolElt, AlgQuatElt, Map`](hilbert-symbols.md#function-hasembedding-fld-algquat) - [`Embed(K, A): Fld, AlgQuat → AlgQuatElt, Map`](hilbert-symbols.md#function-embed-fld-algquat) - [`Embed(Oc, O): RngOrd, AlgAssVOrd → AlgAssVOrdElt, Map`](hilbert-symbols.md#function-embed-rngord-algassvord) - [`Example: Embed`](hilbert-symbols.md#example-ex-06ae1e) - [Predicates on Algebras](predicates.md) - [`IsDefinite(A): AlgQuat → BoolElt`](predicates.md#function-isdefinite-algquat) - [`IsIndefinite(A): AlgQuat → BoolElt`](predicates.md#function-isindefinite-algquat) - [Recognition Functions](matrix-ring-test.md) - [`IsMatrixRing(A): AlgQuat → BoolElt, AlgMat, Map`](matrix-ring-test.md#function-ismatrixring-algquat) - [`MatrixRing(A, eps): AlgQuat, AlgQuatElt → AlgMat, Map`](matrix-ring-test.md#function-matrixring-algquat-algquatelt) - [`MatrixAlgebra(A, eps): AlgQuat, AlgQuatElt → AlgMat, Map`](matrix-ring-test.md#function-matrixalgebra-algquat-algquatelt) - [`Example: Quaternion MatrixRing`](matrix-ring-test.md#example-ex-696aa7) - [`IsQuaternionAlgebra(B): AlgAss → BoolElt, AlgQuat, Map`](matrix-ring-test.md#function-isquaternionalgebra-algass) - [`IsQuaternionAlgebra(B): AlgMat → BoolElt, AlgQuat, Map`](matrix-ring-test.md#function-isquaternionalgebra-algmat) - [`Example: Quaternion IsQuaternionAlgebra`](matrix-ring-test.md#example-ex-52731a) - [`MatrixRepresentation(A): AlgQuat → Map`](matrix-ring-test.md#function-matrixrepresentation-algquat) - [`MatrixRepresentation(R): AlgQuatOrd → Map`](matrix-ring-test.md#function-matrixrepresentation-algquatord) - [Attributes of Orders](order-attributes.md) - [`Algebra(S): AlgQuatOrd → AlgQuat`](order-attributes.md#function-algebra-algquatord) - [`QuaternionAlgebra(S): AlgQuatOrd → AlgQuat`](order-attributes.md#function-quaternionalgebra-algquatord) - [`BasisMatrix(S): AlgQuatOrd → AlgMatElt`](order-attributes.md#function-basismatrix-algquatord) - [`EmbeddingMatrix(S): AlgQuatOrd → AlgMatElt`](order-attributes.md#function-embeddingmatrix-algquatord) - [`Discriminant(S): AlgQuatOrd → RngElt`](order-attributes.md#function-discriminant-algquatord) - [`FactoredDiscriminant(S): AlgQuatOrd → SeqEnum`](order-attributes.md#function-factoreddiscriminant-algquatord) - [`Conductor(S): AlgQuatOrd → RngElt`](order-attributes.md#function-conductor-algquatord) - [`Level(S): AlgQuatOrd → RngElt`](order-attributes.md#function-level-algquatord) - [`Normalizer(S): AlgAssVOrd → Grp, Map`](order-attributes.md#function-normalizer-algassvord) - [Predicates of Orders](order-pred.md) - [`IsMaximal(O): AlgAssVOrd → BoolElt`](order-pred.md#function-ismaximal-algassvord) - [`IspMaximal(O, p): AlgAssVOrd, RngOrdIdl → BoolElt`](order-pred.md#function-ispmaximal-algassvord-rngordidl) - [`IspMaximal(O, p): AlgQuatOrd, RngElt → BoolElt`](order-pred.md#function-ispmaximal-algquatord-rngelt) - [`IsEichler(O): AlgAssVOrd → BoolElt, AlgAssVOrd, AlgAssVOrd`](order-pred.md#function-iseichler-algassvord) - [`IsEichler(O, p): AlgAssVOrd, RngOrdIdl → BoolElt, AlgAssVOrd, AlgAssVOrd`](order-pred.md#function-iseichler-algassvord-rngordidl) - [`IsEichler(O, p): AlgQuatOrd, RngElt → BoolElt, AlgQuatOrd, AlgQuatOrd`](order-pred.md#function-iseichler-algquatord-rngelt) - [`EichlerInvariant(O, p): AlgAssVOrd, RngOrdIdl → RngIntElt`](order-pred.md#function-eichlerinvariant-algassvord-rngordidl) - [`EichlerInvariant(O, p): AlgQuatOrd, RngElt → RngIntElt`](order-pred.md#function-eichlerinvariant-algquatord-rngelt) - [`IsHereditary(O): AlgAssVOrd → BoolElt`](order-pred.md#function-ishereditary-algassvord) - [`IsHereditary(O, p): AlgAssVOrd, RngOrdIdl → BoolElt`](order-pred.md#function-ishereditary-algassvord-rngordidl) - [`IsHereditary(O, p): AlgQuatOrd, RngElt → BoolElt`](order-pred.md#function-ishereditary-algquatord-rngelt) - [`IsGorenstein(O): AlgAssVOrd → BoolElt, .`](order-pred.md#function-isgorenstein-algassvord) - [`IsGorenstein(O, p): AlgAssVOrd, RngOrdIdl → BoolElt, RngIntElt`](order-pred.md#function-isgorenstein-algassvord-rngordidl) - [`IsGorenstein(O, p): AlgQuatOrd, RngElt → BoolElt, RngIntElt`](order-pred.md#function-isgorenstein-algquatord-rngelt) - [`IsBass(O): AlgAssVOrd → BoolElt`](order-pred.md#function-isbass-algassvord) - [`IsBass(O, p): AlgAssVOrd, RngOrdIdl → BoolElt`](order-pred.md#function-isbass-algassvord-rngordidl) - [`IsBass(O, p): AlgQuatOrd, RngElt → BoolElt`](order-pred.md#function-isbass-algquatord-rngelt) - [`IsSameType(O1, O2): AlgAssVOrd, AlgAssVOrd → BoolElt`](order-pred.md#function-issametype-algassvord-algassvord) - [Operations with Orders](order-ops.md) - [`O1 meet O2: AlgQuatOrd[RngInt], AlgQuatOrd[RngInt] → AlgQuatOrd`](order-ops.md#operation-op-meet-algquatord-rngint-algquatord-rngint) - [`O1 meet O2: AlgQuatOrd[RngUPol], AlgQuatOrd[RngUPol] → AlgQuatOrd`](order-ops.md#operation-op-meet-algquatord-rngupol-algquatord-rngupol) - [`O1 meet O2: AlgAssVOrd, AlgAssVOrd → AlgAssVOrd`](order-ops.md#operation-op-meet-algassvord-algassvord) - [`O ^ x: AlgQuatOrd, AlgQuatElt → AlgQuatOrd`](order-ops.md#operation-op-pow-algquatord-algquatelt) - [`O ^ x: AlgAssVOrd, AlgAssElt → AlgAssvOrd`](order-ops.md#operation-op-pow-algassvord-algasselt) - [Ideal Theory of Orders](ideal-theory.md) - [Creation and Access Functions](ideal-theory.md#creation-and-access-functions) - [`LeftIdeal(S, X): AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdl`](ideal-theory.md#function-leftideal-algquatord-algquatelt) - [`lideal: AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdl`](ideal-theory.md#constructor-constructor-lideal-algquatord-algquatelt-algquatordidl) - [`RightIdeal(S, X): AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdl`](ideal-theory.md#function-rightideal-algquatord-algquatelt) - [`rideal: AlgQuatOrd, [AlgQuatElt] → AlgQuatOrd`](ideal-theory.md#constructor-constructor-rideal-algquatord-algquatelt-algquatord) - [`ideal: AlgQuatOrd, [AlgQuatElt] → AlgQuatOrdIdl`](ideal-theory.md#constructor-constructor-ideal-algquatord-algquatelt-algquatordidl) - [`PrimeIdeal(S, p): AlgQuatOrd, RngElt → AlgQuatOrdIdl`](ideal-theory.md#function-primeideal-algquatord-rngelt) - [`CommutatorIdeal(S): AlgQuatOrd → AlgQuatOrdIdl`](ideal-theory.md#function-commutatorideal-algquatord) - [`CommutatorIdeal(S): AlgAssVOrd → AlgAssVOrdIdl`](ideal-theory.md#function-commutatorideal-algassvord) - [`MaximalLeftIdeals(O, p): AlgQuatOrd, RngElt → [AlgQuatOrdIdl]`](ideal-theory.md#function-maximalleftideals-algquatord-rngelt) - [`MaximalLeftIdeals(O, p): AlgAssVOrd, RngOrdIdl → [AlgAssVOrdIdl]`](ideal-theory.md#function-maximalleftideals-algassvord-rngordidl) - [`MaximalRightIdeals(O, p): AlgQuatOrd, RngElt → [AlgQuatOrdIdl]`](ideal-theory.md#function-maximalrightideals-algquatord-rngelt) - [`MaximalRightIdeals(O, p): AlgAssVOrd, RngOrdIdl → [AlgAssVOrdIdl]`](ideal-theory.md#function-maximalrightideals-algassvord-rngordidl) - [`Example: Elementary Ideals`](ideal-theory.md#example-ex-4a8cb2) - [`Example: Ideal Bases`](ideal-theory.md#example-ex-1b3844) - [`LeftOrder(I): AlgQuatOrdIdl → AlgQuatOrd`](ideal-theory.md#function-leftorder-algquatordidl) - [`LeftOrder(I): AlgAssVOrdIdl[RngOrd] → AlgAssVOrd`](ideal-theory.md#function-leftorder-algassvordidl-rngord) - [`RightOrder(I): AlgQuatOrdIdl → AlgQuatOrd`](ideal-theory.md#function-rightorder-algquatordidl) - [`RightOrder(I): AlgAssVOrdIdl[RngOrd] → AlgAssVOrd`](ideal-theory.md#function-rightorder-algassvordidl-rngord) - [`Example: Left Right Quaternion Ordre`](ideal-theory.md#example-ex-c0479d) - [Enumeration of Ideal Classes](ideal-theory.md#enumeration-of-ideal-classes) - [`Mass(S): AlgAssVOrd → FldRatElt`](ideal-theory.md#function-mass-algassvord) - [`LeftIdealClasses(S): AlgQuatOrd → [AlgQuatOrdIdl]`](ideal-theory.md#function-leftidealclasses-algquatord) - [`LeftIdealClasses(S): AlgAssVOrd[RngOrd] → [AlgAssVOrdIdl]`](ideal-theory.md#function-leftidealclasses-algassvord-rngord) - [`RightIdealClasses(S): AlgQuatOrd → [AlgQuatOrdIdl]`](ideal-theory.md#function-rightidealclasses-algquatord) - [`RightIdealClasses(S): AlgAssVOrd[RngOrd] → [AlgAssVOrdIdl]`](ideal-theory.md#function-rightidealclasses-algassvord-rngord) - [`TwoSidedIdealClasses(S): AlgQuatOrd → [AlgQuatOrdIdl]`](ideal-theory.md#function-twosidedidealclasses-algquatord) - [`TwoSidedIdealClasses(S : Support): AlgAssVOrd[RngOrd] → [AlgAssVOrdIdl]`](ideal-theory.md#function-twosidedidealclasses-algassvord-rngord) - [`TwoSidedIdealClassGroup(S : Support): AlgAssVOrd → GrpAb, Map`](ideal-theory.md#function-twosidedidealclassgroup-algassvord) - [`ConjugacyClasses(S): AlgAssVOrd → SeqEnum`](ideal-theory.md#function-conjugacyclasses-algassvord) - [`Example: Ideal Enumeration`](ideal-theory.md#example-ex-092c7e) - [`Example: Ideal Enumeration`](ideal-theory.md#example-ex-81fcc0) - [`Example: Ideal Enumeration`](ideal-theory.md#example-ex-d6b8cf) - [Operations on Ideals](ideal-theory.md#operations-on-ideals) - [`I * J: AlgAssVOrdIdl, AlgAssVOrdIdl → AlgAssVOrdIdl`](ideal-theory.md#operation-op-times-algassvordidl-algassvordidl) - [`I meet J: AlgQuatOrdIdl, AlgQuatOrdIdl → AlgQuatOrdIdl`](ideal-theory.md#operation-op-meet-algquatordidl-algquatordidl) - [`Conjugate(I): AlgQuatOrdIdl → AlgQuatOrdIdl`](ideal-theory.md#function-conjugate-algquatordidl) - [`Conjugate(I): AlgAssVOrdIdl → AlgAssVOrdIdl`](ideal-theory.md#function-conjugate-algassvordidl) - [`Norm(I): AlgQuatOrdIdl → RngElt`](ideal-theory.md#function-norm-algquatordidl) - [`Norm(I): AlgAssVOrdIdl → RngOrdFracIdl`](ideal-theory.md#function-norm-algassvordidl) - [`Factorization(I): AlgQuatOrdIdl → SeqEnum`](ideal-theory.md#function-factorization-algquatordidl) - [`Factorization(I): AlgAssVOrdIdl → SeqEnum`](ideal-theory.md#function-factorization-algassvordidl) - [Norm Spaces and Basis Reduction](norm-space.md) - [`NormSpace(A): AlgQuat → ModTupFld, Map`](norm-space.md#function-normspace-algquat) - [`NormSpace(S): AlgQuatOrd → ModTupRng, Map`](norm-space.md#function-normspace-algquatord) - [`NormModule(S): AlgQuatOrd → ModTupRng, Map`](norm-space.md#function-normmodule-algquatord) - [`GramMatrix(S): AlgQuatOrd → AlgMatElt`](norm-space.md#function-grammatrix-algquatord) - [`GramMatrix(I): AlgQuatOrdIdl → AlgMatElt`](norm-space.md#function-grammatrix-algquatordidl) - [`ReducedGramMatrix(S): AlgQuatOrd[RngInt] → AlgMatElt`](norm-space.md#function-reducedgrammatrix-algquatord-rngint) - [`ReducedGramMatrix(S): AlgQuatOrdIdl[RngInt] → AlgMatElt`](norm-space.md#function-reducedgrammatrix-algquatordidl-rngint) - [`ReducedBasis(S): AlgQuatOrd[RngInt] → SeqEnum`](norm-space.md#function-reducedbasis-algquatord-rngint) - [`ReducedBasis(S): AlgQuatOrdIdl[RngInt] → SeqEnum`](norm-space.md#function-reducedbasis-algquatordidl-rngint) - [`Example: Basis Reduction`](norm-space.md#example-ex-dbe93f) - [`ReducedGramMatrix(S): AlgQuatOrd[RngUPol] → AlgMatElt, SeqEnum`](norm-space.md#function-reducedgrammatrix-algquatord-rngupol) - [`ReducedGramMatrix(S): AlgQuatOrdIdl[RngUPol] → AlgMatElt, SeqEnum`](norm-space.md#function-reducedgrammatrix-algquatordidl-rngupol) - [`ReducedBasis(S): AlgQuatOrd[RngUPol] → SeqEnum`](norm-space.md#function-reducedbasis-algquatord-rngupol) - [`ReducedBasis(S): AlgQuatOrdIdl[RngUPol] → SeqEnum`](norm-space.md#function-reducedbasis-algquatordidl-rngupol) - [`ReducedBasis(O): AlgAssVOrd[RngOrd] → [AlgAssVElt]`](norm-space.md#function-reducedbasis-algassvord-rngord) - [`ReducedBasis(I): AlgAssVOrdIdl[RngOrd] → [AlgAssVOrdElt]`](norm-space.md#function-reducedbasis-algassvordidl-rngord) - [`OptimizedRepresentation(O): AlgAssVOrd → AlgQuat, Map`](norm-space.md#function-optimizedrepresentation-algassvord) - [`OptimisedRepresentation(O): AlgAssVOrd → AlgQuat, Map`](norm-space.md#function-optimisedrepresentation-algassvord) - [`OptimizedRepresentation(A): AlgQuat → AlgQuat, Map`](norm-space.md#function-optimizedrepresentation-algquat) - [`OptimisedRepresentation(A): AlgQuat → AlgQuat, Map`](norm-space.md#function-optimisedrepresentation-algquat) - [`Enumerate(O, A, B): AlgQuatOrd[RngInt], RngIntElt, RngIntElt → [AlgQuatOrdElt]`](norm-space.md#function-enumerate-algquatord-rngint-rngintelt-rngintelt) - [`Enumerate(O, B): AlgQuatOrd[RngInt], RngIntElt → [AlgQuatOrdElt]`](norm-space.md#function-enumerate-algquatord-rngint-rngintelt) - [`Enumerate(O, A, B): AlgQuatOrdIdl[RngInt], RngElt, RngElt → [AlgQuatElt]`](norm-space.md#function-enumerate-algquatordidl-rngint-rngelt-rngelt) - [`Enumerate(O, B): AlgQuatOrd[RngInt], RngElt → [AlgQuatElt]`](norm-space.md#function-enumerate-algquatord-rngint-rngelt) - [`Enumerate(O, A, B): AlgAssVOrd[RngOrd], RngElt, RngElt → [AlgAssVOrdElt]`](norm-space.md#function-enumerate-algassvord-rngord-rngelt-rngelt) - [`Enumerate(O, B): AlgAssVOrd[RngOrd], RngElt → [AlgAssVOrdElt]`](norm-space.md#function-enumerate-algassvord-rngord-rngelt) - [`Enumerate(O, B): AlgAssVOrd[RngOrd], [RngElt] → [AlgAssVOrdElt]`](norm-space.md#function-enumerate-algassvord-rngord-rngelt-2) - [`Enumerate(I, B): AlgAssVOrdIdl[RngOrd], [RngElt] → [AlgAssVOrdElt]`](norm-space.md#function-enumerate-algassvordidl-rngord-rngelt) - [Isomorphisms](isomorphisms.md) - [Isomorphisms of Algebras](isomorphisms.md#isomorphisms-of-algebras) - [`IsIsomorphic(A, B): AlgQuat, AlgQuat → BoolElt, Map`](isomorphisms.md#function-isisomorphic-algquat-algquat) - [Isomorphisms of Orders](isomorphisms.md#isomorphisms-of-orders) - [`IsIsomorphic(S, T): AlgQuatOrd, AlgQuatOrd → BoolElt, Map, AlgQuatElt`](isomorphisms.md#function-isisomorphic-algquatord-algquatord) - [`IsIsomorphic(S, T): AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] → BoolElt, Map, AlgQuatElt`](isomorphisms.md#function-isisomorphic-algassvord-rngord-algassvord-rngord) - [`IsConjugate(S, T): AlgAssVOrd, AlgAssVOrd → BoolElt, Map, AlgQuatElt`](isomorphisms.md#function-isconjugate-algassvord-algassvord) - [`Isomorphism(S, T): AlgQuatOrd, AlgQuatOrd → Map`](isomorphisms.md#function-isomorphism-algquatord-algquatord) - [Isomorphisms of Ideals](isomorphisms.md#isomorphisms-of-ideals) - [`IsIsomorphic(I, J): AlgAssVOrdIdl, AlgAssVOrdIdl → BoolElt, AlgAssVElt`](isomorphisms.md#function-isisomorphic-algassvordidl-algassvordidl) - [`IsPrincipal(I): AlgAssVOrdIdl → BoolElt, AlgQuatElt`](isomorphisms.md#function-isprincipal-algassvordidl) - [`IsLeftIsomorphic(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → BoolElt, Map, AlgQuatElt`](isomorphisms.md#function-isleftisomorphic-algquatordidl-algquatordidl) - [`IsRightIsomorphic(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → BoolElt, Map, AlgQuatElt`](isomorphisms.md#function-isrightisomorphic-algquatordidl-algquatordidl) - [`IsLeftIsomorphic(I, J): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] → BoolElt, AlgQuatElt`](isomorphisms.md#function-isleftisomorphic-algassvordidl-rngord-algassvordidl-rngord) - [`IsRightIsomorphic(I, J): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] → BoolElt, AlgQuatElt`](isomorphisms.md#function-isrightisomorphic-algassvordidl-rngord-algassvordidl-rngord) - [`LeftIsomorphism(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → Map, AlgQuatElt`](isomorphisms.md#function-leftisomorphism-algquatordidl-algquatordidl) - [`RightIsomorphism(I, J): AlgQuatOrdIdl, AlgQuatOrdIdl → Map, AlgQuatElt`](isomorphisms.md#function-rightisomorphism-algquatordidl-algquatordidl) - [Examples](isomorphisms.md#examples) - [`Example: Isomorphism Algebras`](isomorphisms.md#example-ex-f5c0d5) - [`Example: Isomorphism Example`](isomorphisms.md#example-ex-50db58) - [`Example: Left Right Isomorphisms`](isomorphisms.md#example-ex-03d026) - [`Example: Left Right Isomorphisms Number Field`](isomorphisms.md#example-ex-0c04cb) - [Units and Unit Groups](unit-group.md) - [`NormOneGroup(S): AlgAssVOrd → GrpPerm, Map`](unit-group.md#function-normonegroup-algassvord) - [`NormOneGroup(S): AlgQuatOrd → GrpAb, Map`](unit-group.md#function-normonegroup-algquatord) - [`Units(S): AlgQuatOrd → SeqEnum`](unit-group.md#function-units-algquatord) - [`Units(S): AlgAssVOrd → SeqEnum`](unit-group.md#function-units-algassvord) - [`MultiplicativeGroup(S): AlgQuatOrd[RngInt] → GrpPerm, Map`](unit-group.md#function-multiplicativegroup-algquatord-rngint) - [`UnitGroup(S): AlgQuatOrd[RngInt] → GrpPerm, Map`](unit-group.md#function-unitgroup-algquatord-rngint) - [`MultiplicativeGroup(S): AlgQuatOrd[RngUPol] → GrpAb, Map`](unit-group.md#function-multiplicativegroup-algquatord-rngupol) - [`UnitGroup(S): AlgQuatOrd[RngUPol] → GrpAb, Map`](unit-group.md#function-unitgroup-algquatord-rngupol) - [`MultiplicativeGroup(S): AlgAssVOrd[RngOrd] → GrpPerm, Map`](unit-group.md#function-multiplicativegroup-algassvord-rngord) - [`UnitGroup(S): AlgAssVOrd[RngOrd] → GrpPerm, Map`](unit-group.md#function-unitgroup-algassvord-rngord) - [`Example: Unit Group`](unit-group.md#example-ex-410b55) - [`Example: Unit Group NumberRing`](unit-group.md#example-ex-b816c0)