Introduction

 
Construction
      EtaleAlgebra(seq) : SeqEnum[FldNum] -> AlgEtQ
      Example AlgEtQ_TwoCopiesOfQ (H42E1)
      EtaleAlgebra(f) : RngUPolElt[RngInt] -> AlgEtQ
      DirectProduct(seq) : SeqEnum[AlgEtQ] -> AlgEtQ, SeqEnum[Map], SeqEnum[Map]

 
Attributes
      DefiningPolynomial(A) : AlgEtQ -> RngUPolElt
      Components(A) : AlgEtQ -> SeqEnum
      Dimension(A) : AlgEtQ -> RngInt
      AbsoluteDimension(A) : AlgEtQ -> RngInt
      HasBaseField(A) : AlgEtQ -> BoolElt
      BaseField(A) : AlgEtQ -> FldNum, Map
      PrimeField(A) : AlgEtQ -> FldNum
      IsNumberField(A) : AlgEtQ -> BoolElt, FldNum, Map
      A1 eq A2 : AlgEtQ, AlgEtQ -> BoolElt

 
Homomorphisms
      HomsToC(A) : AlgEtQ -> SeqEnum[Map]
      Example AlgEtQ_HomsToCExample (H42E2)
      Hom(A, B, img) : AlgEtQ, AlgEtQ, SeqEnum[AlgEtQElt] -> Map
      DiagonalEmbedding(K, V) : AlgEtQ, AlgEtQ -> Map
      Example AlgEtQ_HomAndDiagonal (H42E3)

 
Elements
      Parent(x) : AlgEtQElt -> AlgEtQ
      Components(x) : AlgEtQElt -> SeqEnum
      AbsoluteCoordinates(x) : AlgEtQElt -> SeqEnum
      AbsoluteCoordinates(x, S) : AlgEtQElt, AlgEtQOrd -> SeqEnum
      IsCoercible(A, x) : AlgEtQ, Any -> BoolElt, AlgEtQElt
      A ! x : AlgEtQ, Any) -> AlgEtQElt
      One(A) : AlgEtQ -> AlgEtQElt
      Zero(A) : AlgEtQ -> AlgEtQElt
      IsUnit(x) : AlgEtQElt -> BoolElt
      IsZeroDivisor(x) : AlgEtQElt -> BoolElt
      Random(A, bd) : AlgEtQ, RngIntElt -> AlgEtQElt
      Random(A) : AlgEtQ -> AlgEtQElt
      RandomUnit(A, bd) : AlgEtQ, RngIntElt -> AlgEtQElt
      x1 eq x2 : AlgEtQElt, AlgEtQElt -> BoolElt
      Inverse(x) : AlgEtQElt -> AlgEtQElt
      &+ seq : SeqEnum[AlgEtQElt] -> AlgEtQElt
      &* seq : SeqEnum[AlgEtQElt] -> AlgEtQElt
      DotProduct(a, b) : SeqEnum, SeqEnum -> Any
      Example AlgEtQ_DotProductExample (H42E4)
      MinimalPolynomial(x) : AlgEtQElt -> RngUPolElt
      MinimalPolynomial(x, F) : AlgEtQElt, Rng -> RngUPolElt
      AbsoluteMinimalPolynomial(x) : AlgEtQElt -> RngUPolElt
      IsIntegral(x) : AlgEtQElt -> BoolElt
      Evaluate(f, a) : RngUPolElt, AlgEtQElt -> AlgEtQElt
      PrimitiveElement(A) : AlgEtQ -> AlgEtQElt
      PowerBasis(A) : AlgEtQ -> SeqEnum[AlgEtQElt]
      Basis(A) : AlgEtQ -> SeqEnum
      AbsoluteBasis(A) : AlgEtQ -> SeqEnum
      A . i : AlgEtQ, RngIntElt -> AlgEtQElt
      AbsoluteCoordinates(seq, basis) : SeqEnum[AlgEtQElt], SeqEnum[AlgEtQElt] -> SeqEnum
      OrthogonalIdempotents(A) : AlgEtQ -> SeqEnum
      Idempotents(A) : AlgEtQ -> SeqEnum

 
Orders of Algebras
      IsCoercible(S, x) : AlgEtQOrd, Any -> BoolElt, AlgEtQElt
      Order(gens) : SeqEnum[AlgEtQElt] -> AlgEtQOrd
      Order(A, orders) : AlgEtQ, Tup -> AlgEtQOrd
      Algebra(S) : AlgEtQOrd -> AlgEtQ
      ZBasis(S) : AlgEtQOrd -> SeqEnum[AlgEtQElt]
      Generators(S) : AlgEtQOrd ->SeqEnum[AlgEtQElt]
      O1 eq O2 : AlgEtQOrd, AlgEtQOrd -> BoolElt
      x in O : AlgEtQElt, AlgEtQOrd -> BoolElt
      AbsoluteCoordinates(seq, O) : SeqEnum[AlgEtQElt], AlgEtQOrd -> SeqEnum
      One(S) : AlgEtQOrd -> AlgEtQElt
      Zero(S) : AlgEtQOrd -> AlgEtQElt
      Random(O, bd) : AlgEtQOrd, RngIntElt -> AlgEtQElt
      Random(O) : AlgEtQOrd -> AlgEtQElt
      IsKnownOrder(~R) : AlgEtQOrd ->
      EquationOrder(A) : AlgEtQ -> AlgEtQOrd
      ProductOfEquationOrders(A) : AlgEtQ -> AlgEtQOrd
      MaximalOrder(A) : AlgEtQ -> AlgEtQOrd
      IsMaximal(S) : AlgEtQOrd -> BoolElt
      IsProductOfOrders(O) : AlgEtQOrd -> BoolElt, Tup
      IsProductOfOrdersInComponents(O) : AlgEtQOrd -> BoolElt, Tup
      IsProductOfOrdersInFactorAlgebras(S) : AlgEtQOrd -> BoolElt, SeqEnum[AlgEtQElt]
      Example AlgEtQ_OrdersFactorAlgebras (H42E5)
      Index(T) : AlgEtQOrd -> FldRatElt
      Index(S, T) : AlgEtQOrd, AlgEtQOrd -> FldRatElt
      O1 subset O2 : AlgEtQOrd, AlgEtQOrd -> BoolElt
      O1 * O2 : AlgEtQOrd, AlgEtQOrd -> AlgEtQOrd
      O1 meet O2 : AlgEtQOrd, AlgEtQOrd -> AlgEtQOrd
      MultiplicatorRing(R) : AlgEtQOrd -> AlgEtQOrd

 
Ideals
      Ideal(S, gens) : AlgEtQOrd, SeqEnum -> AlgEtQIdl
      Ideal(S, idls) : AlgEtQOrd, Tup -> AlgEtQIdl
      Ideal(S, gen) : AlgEtQOrd, Any -> AlgEtQIdl
      T !! I : AlgEtQOrd, AlgEtQIdl -> AlgEtQIdl
      Algebra(I) : AlgEtQIdl -> AlgEtQ
      Order(I) : AlgEtQIdl -> AlgEtQOrd
      ZBasis(I) : AlgEtQIdl -> SeqEnum[AlgEtQElt]
      Generators(I) : AlgEtQIdl -> SeqEnum[AlgEtQElt]
      I eq J : AlgEtQIdl, AlgEtQIdl -> BoolElt
      I eq S : AlgEtQIdl, AlgEtQOrd -> BoolElt
      AbsoluteCoordinates(x, I) : AlgEtQElt, AlgEtQIdl -> SeqEnum
      AbsoluteCoordinates(seq, I) : SeqEnum[AlgEtQElt], AlgEtQIdl -> SeqEnum
      x in I : AlgEtQElt, AlgEtQIdl -> BoolElt
      S subset I : AlgEtQOrd, AlgEtQIdl -> BoolElt
      I subset S : AlgEtQIdl, AlgEtQOrd -> BoolElt
      I1 subset I2 : AlgEtQIdl, AlgEtQIdl -> BoolElt
      Index(T) : AlgEtQIdl -> FldRatElt
      Index(J, I) : AlgEtQIdl, AlgEtQIdl -> Any
      Index(S, I) : AlgEtQOrd, AlgEtQIdl -> Any
      OneIdeal(S) : AlgEtQOrd -> AlgEtQIdl
      Conductor(O) : AlgEtQOrd -> AlgEtQOrdIdl
      I + J : AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl
      I * J : AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl
      I * x : AlgEtQIdl, AlgEtQElt -> AlgEtQIdl
      I ^ n : AlgEtQIdl, RngIntElt) -> AlgEtQIdl
      I meet S : AlgEtQIdl, AlgEtQOrd -> AlgEtQIdl
      I meet J : AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl
      &+ seq : SeqEnum[AlgEtQIdl] -> AlgEtQIdl
      ColonIdeal(I, J) : AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl
      ColonIdeal(O, J) : AlgEtQOrd, AlgEtQIdl -> AlgEtQIdl
      ColonIdeal(I, O) : AlgEtQIdl, AlgEtQOrd -> AlgEtQIdl
      IsInvertible(I) : AlgEtQIdl -> BoolElt
      Inverse(I) : AlgEtQIdl -> AlgEtQIdl
      MultiplicatorRing(I) : AlgEtQIdl -> AlgEtQOrd
      IsProductOfIdeals(I) : AlgEtQIdl -> BoolElt, Tup
      Random(I, bd) : AlgEtQIdl, RngIntElt -> AlgEtQElt
      Random(I) : AlgEtQIdl -> AlgEtQElt
      IsCoprime(I, J) : AlgEtQIdl, AlgEtQIdl -> BoolElt
      IsIntegral(I) : AlgEtQIdl -> BoolElt
      MakeIntegral(I) : AlgEtQIdl -> AlgEtQIdl, RngIntElt
      MinimalInteger(I) : AlgEtQIdl -> RngIntElt
      CoprimeRepresentative(I, J) : AlgEtQIdl, AlgEtQIdl -> AlgEtQElt, AlgEtQIdl
      ZBasisLLL(~S) : AlgEtQOrd ->

 
Quotients
      Quotient(I, zbJ) : AlgEtQIdl, SeqEnum[AlgEtQElt] -> GrpAb, Map
      Quotient(I, J) : AlgEtQIdl, AlgEtQIdl -> GrpAb, Map
      Quotient(S, zbJ) : AlgEtQOrd, SeqEnum[AlgEtQElt] -> GrpAb, Map
      ResidueRing(S, I) : AlgEtQOrd, AlgEtQIdl -> GrpAb , Map
      ResidueField(P) : AlgEtQIdl -> FldFin, Map
      PrimitiveElementResidueField(P) : AlgEtQIdl->AlgEtQElt
      QuotientVS(I, J, P) : AlgEtQOrd, AlgEtQOrd, AlgEtQIdl -> ModRng, Map
      Example AlgEtQ_QuotientsResidues (H42E6)
      QuotientVS(I, J, P) : AlgEtQOrd, AlgEtQIdl, AlgEtQIdl -> ModRng, Map
      QuotientVS(I, J, P) : AlgEtQIdl, AlgEtQOrd, AlgEtQIdl -> ModRng, Map
      QuotientVS(I, J, P) : AlgEtQIdl, AlgEtQIdl, AlgEtQIdl -> ModRng, Map

 
Over Orders
      IsMaximalAtPrime(R, P) : AlgEtQOrd, AlgEtQIdl -> BoolElt
      MinimalOverOrdersAtPrime(R, P) : AlgEtQOrd, AlgEtQIdl -> SetIndx[AlgEtQOrd]
      MinimalOverOrders(R) : AlgEtQOrd -> SetIndx[AlgEtQOrd]
      OverOrdersAtPrime(R, P) : AlgEtQOrd, AlgEtQIdl -> SeqEnum[AlgEtQOrd]
      OverOrders(R) : AlgEtQOrd -> SeqEnum[AlgEtQOrd]
      Example AlgEtQ_OverOrdersExample (H42E7)
      FindOverOrders(R) : AlgEtQOrd -> SetIndx[AlgEtQOrd]

 
Over Order Graph
      GraphOverOrders(R) : AlgEtQOrd -> GrphDir

 
Trace and Norm
      Trace(x) : AlgEtQElt -> FldRatElt
      Norm(x) : AlgEtQElt -> FldRatElt
      AbsoluteTrace(x) : AlgEtQElt -> FldRatElt
      AbsoluteNorm(x) : AlgEtQElt -> FldRatElt
      TraceDualIdeal(I) : AlgEtQIdl -> AlgEtQIdl
      TraceDualIdeal(O) : AlgEtQOrd -> AlgEtQIdl

 
Completion
      Completion(P) : AlgEtQIdl -> FldPad,Map
      Example AlgEtQ_UniformizersCompletion (H42E8)

 
Intermediate Ideals
      IntermediateIdeals(I, J) : AlgEtQIdl, AlgEtQIdl -> SetIndx[AlgEtQIdl]
      IntermediateIdeals(I, J, O) :AlgEtQIdl, AlgEtQIdl, AlgEtQOrd -> SetIndx[AlgEtQIdl]
      IntermediateIdeals(I, J, N) : AlgEtQIdl, AlgEtQIdl, RngIntElt->SetIndx[AlgEtQIdl]

 
Ideals of Index
      IdealsOfIndex(O, N) : RngOrd, RngIntElt -> SeqEnum[RngOrdIdl]
      IdealsOfIndex(I, N) : RngOrdIdl, RngIntElt -> SeqEnum[RngOrdIdl]
      IdealsOfIndex(I, N) : RngOrdFracIdl, RngIntElt -> SeqEnum[RngOrdFracIdl]
      IdealsOfIndex(I, N) : AlgEtQIdl, RngIntElt -> SeqEnum[AlgEtQIdl]
      IdealsOfIndex(O, N) : AlgEtQOrd, RngIntElt -> SeqEnum[AlgEtQIdl]

 
Short Element and Small Representative
      ShortElement(I) : AlgEtQIdl ->AlgEtQElt
      SmallRepresentative(I) : AlgEtQIdl ->AlgEtQIdl,AlgEtQElt

 
Minimal Generators
      TwoGeneratingSet(I) : AlgEtQIdl ->

 
Chinese Remainder Theorem
      ChineseRemainderTheorem(Is, as) : SeqEnum[AlgEtQIdl], SeqEnum[AlgEtQElt]-> AlgEtQElt
      ChineseRemainderTheorem(I, J, a, b) : AlgEtQIdl, AlgEtQIdl, AlgEtQElt, AlgEtQElt -> AlgEtQElt
      ChineseRemainderTheoremFunctions(Is) : SeqEnum[AlgEtQIdl] -> Map, Map
      Example AlgEtQ_CRTFunctions (H42E9)

 
Picard Group
      ResidueRingUnits(S, I) : AlgEtQOrd, AlgEtQIdl -> GrpAb,Map
      ResidueRingUnits(I) : AlgEtQIdl -> GrpAb,Map
      ResidueRingUnitsSubgroupGenerators(F) : AlgEtQIdl -> SeqEnum[AlgEtQElt]
      IsPrincipal(I1) : AlgEtQIdl ->BoolElt, AlgAssElt
      PicardGroup(S) : AlgEtQOrd -> GrpAb, Map
      ExtensionHomPicardGroups(S, T) : AlgEtQOrd, AlgEtQOrd -> Map
      UnitGroup(S) : AlgEtQOrd -> GrpAb, Map
      Example AlgEtQ_PicardAndUnits (H42E10)
      IsIsomorphic(I, J) : AlgEtQIdl, AlgEtQIdl -> BoolElt, AlgAssElt

 
Factorization and Primes
      Factorization(I) : AlgEtQIdl -> Tup
      PrimesAbove(I) : AlgEtQIdl -> SeqEnum[AlgAssEtOrdIdl]
      SingularPrimes(R) : AlgEtQOrd -> SeqEnum[AlgAssEtOrdIdl]
      PlacesAboveRationalPrime(E, p) : AlgEtQ, RngIntElt -> SeqEnum[AlgEtQIdl]
      NonInvertiblePrimes(R) : AlgEtQOrd -> SetIndx
      IsPrime(I) : AlgEtQIdl -> BoolElt
      Valuation(x, P) : AlgEtQElt, AlgEtQIdl -> RngIntElt
      InertiaDegree(P) : AlgEtQIdl -> RngIntElt
      IsBassAtPrime(S, P) : AlgEtQOrd, AlgEtQIdl -> BoolElt
      IsBass(S) : AlgEtQOrd -> BoolElt
      IsGorensteinAtPrime(S, P) : AlgEtQOrd, AlgEtQIdl -> BoolElt
      IsGorenstein(O) : AlgEtQOrd->BoolElt
      Uniformizers(PPs) : SeqEnum[AlgEtQIdl] -> SeqEnum

 
Low Cohen Macauley Type
      NonGorensteinPrimes(S) : AlgEtQOrd->SeqEnum,SeqEnum
      CohenMacaulayTypeAtPrime(S, P) : AlgEtQOrd, AlgEtQIdl->RngIntElt
      CohenMacaulayType(S) : AlgEtQOrd->RngIntElt
      Example AlgEtQ_MaxCohenMacaulayType (H42E11)

 
Weak Classes
      WKICM_bar(S) : AlgEtQOrd -> SeqEnum
      WeakEquivalenceClassesWithPrescribedMultiplicatorRing(S) : AlgEtQOrd -> SeqEnum[AlgEtQIdl]
      WeakEquivalenceClassMonoid(E) : AlgEtQOrd -> SeqEnum[AlgEtQIdl]
      WeakEquivalenceClassMonoidAbstract(R) : AlgEtQOrd -> AlgEtQWECM,Map
      W ! x : AlgEtQWECM, Any -> AlgEtQWECMElt
      x in W : AlgEtQWECMElt, AlgEtQWECM -> BoolElt
      Parent(x) : AlgEtQWECMElt -> AlgEtQWECM
      MultiplicationTable(W) : AlgEtQWECM -> Assoc
      Example AlgEtQ_WeakEquivalenceClasses (H42E12)

 
Weak Testing
      IsWeakEquivalent(I, J) : AlgEtQIdl, AlgEtQIdl->BoolElt
      IsWeakEquivalent(O1, O2) : AlgEtQOrd, AlgEtQOrd->BoolElt
      IsWeakEquivalent(O, J) : AlgEtQOrd, AlgEtQIdl->BoolElt
      Example AlgEtQ_WeakTesting (H42E13)
      IsWeakEquivalent(J, O) : AlgEtQIdl, AlgEtQOrd->BoolElt

 
Ideal Class Monoid
      ICM_bar(S) : AlgEtQOrd -> SeqEnum
      ICM(S) : AlgEtQOrd -> SeqEnum
      IdealClassMonoidAbstract(R) : AlgEtQOrd -> AlgEtQICM,Map
      icm ! x : AlgEtQICM, Any -> AlgEtQICMElt
      x in icm : AlgEtQICMElt, AlgEtQICM -> BoolElt

 
Complex Conjugation
      HasComplexConjugate(A) : AlgEtQ -> BoolElt
      ComplexConjugate(x) : AlgEtQElt -> AlgEtQElt
      IsConjugateStable(O) : AlgEtQOrd -> BoolElt,AlgEtQOrd
      ComplexConjugate(O) : AlgEtQOrd -> AlgEtQOrd
      IsConjugateStable(I) : AlgEtQIdl -> BoolElt,AlgEtQIdl
      ComplexConjugate(I) : AlgEtQIdl -> AlgEtQIdl

 
Complex Multiplication
      CMType(seq) : SeqEnum[Map] -> AlgEtQCMType
      Example AlgEtQ_CMTypes (H42E14)
      CreateCMType(seq) : SeqEnum[Map] -> AlgEtQCMType
      CMType(b) : AlgEtQElt -> AlgEtQCMType
      CreateCMType(b) : AlgEtQElt -> AlgEtQCMType
      CMPositiveElement(PHI) : AlgEtQCMType ->AlgEtQElt
      CMPosElt(PHI) : AlgEtQCMType ->AlgEtQElt
      Homs(PHI) : AlgEtQCMType ->SeqEnum[Map]
      PHI1 eq PHI2 : AlgEtQCMType, AlgEtQCMType ->BoolElt
      Precision(PHI) : AlgEtQCMType->RngIntElt
      ChangePrecision(PHI0, prec) : AlgEtQCMType, RngIntElt ->AlgEtQCMType
      ChangePrecision(~PHI, prec) : AlgEtQCMType, RngIntElt ->
      AllCMTypes(A) : AlgEtQ -> SeqEnum[AlgEtQCMType]

 
Totally Real and Positive
      IsTotallyReal(a) : AlgEtQElt -> BoolElt
      IsTotallyRealPositive(a) : AlgEtQElt -> BoolElt
      TotallyRealSubAlgebra(K) : AlgEtQ -> AlgEtQ,Map
      TotallyRealUnitGroup(S) : AlgEtQOrd -> Grp
      TotallyRealPositiveUnitGroup(S) : AlgEtQOrd -> Grp

 
Printing and Saving
      PrintSeqAlgEtQElt(seq) : SeqEnum[AlgEtQElt] -> SeqEnum, MonStgElt
      PrintWKICM(R) : AlgEtQOrd -> MonStgElt
      LoadWKICM(str) : MonStgElt -> AlgEtQOrd

 
Bibliography

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