Étale Algebras
- Introduction
- Construction
- Attributes
- Homomorphisms
- Elements
Parent(x): AlgEtQElt → AlgEtQAlgebra(x): AlgEtQElt → AlgEtQComponents(x): AlgEtQElt → SeqEnumAbsoluteCoordinates(x): AlgEtQElt → SeqEnumAbsoluteCoordinates(x, S): AlgEtQElt, AlgEtQOrd → SeqEnumIsCoercible(A, x): AlgEtQ, Any → BoolElt, AlgEtQEltA ! x: AlgEtQ, Any → AlgEtQEltOne(A): AlgEtQ → AlgEtQEltZero(A): AlgEtQ → AlgEtQEltIsUnit(x): AlgEtQElt → BoolEltIsZeroDivisor(x): AlgEtQElt → BoolEltRandom(A, bd): AlgEtQ, RngIntElt → AlgEtQEltRandom(A): AlgEtQ → AlgEtQEltRandomUnit(A, bd): AlgEtQ, RngIntElt → AlgEtQEltx1 eq x2: AlgEtQElt, AlgEtQElt → BoolEltx1 eq x2: RngIntElt, AlgEtQElt → BoolEltx1 eq x2: AlgEtQElt, RngIntElt → BoolEltx1 eq x2: FldRatElt, AlgEtQElt → BoolEltx1 eq x2: AlgEtQElt, FldRatElt → BoolEltx1 + x2: AlgEtQElt, AlgEtQElt → AlgEtQEltx1 + x2: Any, AlgEtQElt → AlgEtQEltx1 + x2: AlgEtQElt, Any → AlgEtQElt- x: AlgEtQElt → AlgEtQEltx1 - x2: AlgEtQElt, AlgEtQElt → AlgEtQEltx1 - x2: Any, AlgEtQElt → AlgEtQEltx1 - x2: AlgEtQElt, Any → AlgEtQEltx1 * x2: AlgEtQElt, AlgEtQElt → AlgEtQEltx1 * x2: Any, AlgEtQElt → AlgEtQEltx1 * x2: AlgEtQElt, Any → AlgEtQEltInverse(x): AlgEtQElt → AlgEtQEltx ^ n: AlgEtQElt, RngIntElt → AlgEtQEltx1 / x2: AlgEtQElt, AlgEtQElt → AlgEtQEltx1 / x2: Any, AlgEtQElt → AlgEtQEltx1 / x2: AlgEtQElt, Any → AlgEtQElt&+ seq: SeqEnum[AlgEtQElt] → AlgEtQElt&* seq: SeqEnum[AlgEtQElt] → AlgEtQEltDotProduct(a, b): SeqEnum, SeqEnum → Any- Example: Components and Idempotents of an Etale Algebra
MinimalPolynomial(x): AlgEtQElt → RngUPolEltMinimalPolynomial(x, F): AlgEtQElt, Rng → RngUPolEltAbsoluteMinimalPolynomial(x): AlgEtQElt → RngUPolEltIsIntegral(x): AlgEtQElt → BoolEltEvaluate(f, a): RngUPolElt, AlgEtQElt → AlgEtQEltPrimitiveElement(A): AlgEtQ → AlgEtQEltPowerBasis(A): AlgEtQ → SeqEnum[AlgEtQElt]Basis(A): AlgEtQ → SeqEnumAbsoluteBasis(A): AlgEtQ → SeqEnumA . i: AlgEtQ, RngIntElt → AlgEtQEltAbsoluteCoordinates(seq, basis): SeqEnum[AlgEtQElt], SeqEnum[AlgEtQElt] → SeqEnumOrthogonalIdempotents(A): AlgEtQ → SeqEnumIdempotents(A): AlgEtQ → SeqEnum
- Orders of Algebras
IsCoercible(S, x): AlgEtQOrd, Any → BoolElt, AlgEtQEltOrder(gens): SeqEnum[AlgEtQElt] → AlgEtQOrdOrder(A, orders): AlgEtQ, Tup → AlgEtQOrdAlgebra(S): AlgEtQOrd → AlgEtQZBasis(S): AlgEtQOrd → SeqEnum[AlgEtQElt]Generators(S): AlgEtQOrd → SeqEnum[AlgEtQElt]O1 eq O2: AlgEtQOrd, AlgEtQOrd → BoolEltx in O: AlgEtQElt, AlgEtQOrd → BoolEltx in O: RngIntElt, AlgEtQOrd → BoolEltx in O: FldRatElt, AlgEtQOrd → BoolEltAbsoluteCoordinates(seq, O): SeqEnum[AlgEtQElt], AlgEtQOrd → SeqEnumOne(S): AlgEtQOrd → AlgEtQEltZero(S): AlgEtQOrd → AlgEtQEltRandom(O, bd): AlgEtQOrd, RngIntElt → AlgEtQEltRandom(O): AlgEtQOrd → AlgEtQEltIsKnownOrder(~R): AlgEtQOrdEquationOrder(A): AlgEtQ → AlgEtQOrdProductOfEquationOrders(A): AlgEtQ → AlgEtQOrdMaximalOrder(A): AlgEtQ → AlgEtQOrdIsMaximal(S): AlgEtQOrd → BoolEltIsProductOfOrders(O): AlgEtQOrd → BoolElt, TupIsProductOfOrdersInComponents(O): AlgEtQOrd → BoolElt, TupIsProductOfOrdersInFactorAlgebras(S): AlgEtQOrd → BoolElt, SeqEnum[AlgEtQElt]- Example: Orders in a Product of Number Fields
Index(T): AlgEtQOrd → FldRatEltIndex(S, T): AlgEtQOrd, AlgEtQOrd → FldRatEltO1 subset O2: AlgEtQOrd, AlgEtQOrd → BoolEltO1 * O2: AlgEtQOrd, AlgEtQOrd → AlgEtQOrdO1 meet O2: AlgEtQOrd, AlgEtQOrd → AlgEtQOrdMultiplicatorRing(R): AlgEtQOrd → AlgEtQOrd
- Ideals
Ideal(S, gens): AlgEtQOrd, SeqEnum → AlgEtQIdlIdeal(S, idls): AlgEtQOrd, Tup → AlgEtQIdlIdeal(S, gen): AlgEtQOrd, Any → AlgEtQIdlS * gen: AlgEtQOrd, AlgEtQElt → AlgEtQIdlS * gen: AlgEtQOrd, RngIntElt → AlgEtQIdlS * gen: AlgEtQOrd, FldRatElt → AlgEtQIdlgen * S: AlgEtQElt, AlgEtQOrd → AlgEtQIdlgen * S: RngIntElt, AlgEtQOrd → AlgEtQIdlgen * S: FldRatElt, AlgEtQOrd → AlgEtQIdlT !! I: AlgEtQOrd, AlgEtQIdl → AlgEtQIdlAlgebra(I): AlgEtQIdl → AlgEtQOrder(I): AlgEtQIdl → AlgEtQOrdZBasis(I): AlgEtQIdl → SeqEnum[AlgEtQElt]Generators(I): AlgEtQIdl → SeqEnum[AlgEtQElt]I eq J: AlgEtQIdl, AlgEtQIdl → BoolEltI ne J: AlgEtQIdl, AlgEtQIdl → BoolEltI eq S: AlgEtQIdl, AlgEtQOrd → BoolEltS eq I: AlgEtQOrd, AlgEtQIdl → BoolEltAbsoluteCoordinates(x, I): AlgEtQElt, AlgEtQIdl → SeqEnumAbsoluteCoordinates(seq, I): SeqEnum[AlgEtQElt], AlgEtQIdl → SeqEnumx in I: AlgEtQElt, AlgEtQIdl → BoolEltx in I: RngIntElt, AlgEtQIdl → BoolEltx in I: FldRatElt, AlgEtQIdl → BoolEltS subset I: AlgEtQOrd, AlgEtQIdl → BoolEltI subset S: AlgEtQIdl, AlgEtQOrd → BoolEltI1 subset I2: AlgEtQIdl, AlgEtQIdl → BoolEltIndex(T): AlgEtQIdl → FldRatEltIndex(J, I): AlgEtQIdl, AlgEtQIdl → AnyIndex(S, I): AlgEtQOrd, AlgEtQIdl → AnyOneIdeal(S): AlgEtQOrd → AlgEtQIdlConductor(O): AlgEtQOrd → AlgEtQOrdIdlI + J: AlgEtQIdl, AlgEtQIdl → AlgEtQIdlI * J: AlgEtQIdl, AlgEtQIdl → AlgEtQIdlI * x: AlgEtQIdl, AlgEtQElt → AlgEtQIdlI * x: AlgEtQIdl, RngIntElt → AlgEtQIdlI * x: AlgEtQIdl, FldRatElt → AlgEtQIdlx * I: AlgEtQElt, AlgEtQIdl → AlgEtQIdlx * I: RngIntElt, AlgEtQIdl → AlgEtQIdlx * I: FldRatElt, AlgEtQIdl → AlgEtQIdlI ^ n: AlgEtQIdl, RngIntElt → AlgEtQIdlI meet S: AlgEtQIdl, AlgEtQOrd → AlgEtQIdlS meet I: AlgEtQOrd, AlgEtQIdl → AlgEtQIdlI meet J: AlgEtQIdl, AlgEtQIdl → AlgEtQIdl&+ seq: SeqEnum[AlgEtQIdl] → AlgEtQIdlColonIdeal(I, J): AlgEtQIdl, AlgEtQIdl → AlgEtQIdlColonIdeal(O, J): AlgEtQOrd, AlgEtQIdl → AlgEtQIdlColonIdeal(I, O): AlgEtQIdl, AlgEtQOrd → AlgEtQIdlIsInvertible(I): AlgEtQIdl → BoolEltInverse(I): AlgEtQIdl → AlgEtQIdlMultiplicatorRing(I): AlgEtQIdl → AlgEtQOrdIsProductOfIdeals(I): AlgEtQIdl → BoolElt, TupRandom(I, bd): AlgEtQIdl, RngIntElt → AlgEtQEltRandom(I): AlgEtQIdl → AlgEtQEltIsCoprime(I, J): AlgEtQIdl, AlgEtQIdl → BoolEltIsIntegral(I): AlgEtQIdl → BoolEltMakeIntegral(I): AlgEtQIdl → AlgEtQIdl, RngIntEltMinimalInteger(I): AlgEtQIdl → RngIntEltCoprimeRepresentative(I, J): AlgEtQIdl, AlgEtQIdl → AlgEtQElt, AlgEtQIdlZBasisLLL(~S): AlgEtQOrdZBasisLLL(~S): AlgEtQIdl
- Quotients
Quotient(I, zbJ): AlgEtQIdl, SeqEnum[AlgEtQElt] → GrpAb, MapQuotient(I, J): AlgEtQIdl, AlgEtQIdl → GrpAb, MapQuotient(S, zbJ): AlgEtQOrd, SeqEnum[AlgEtQElt] → GrpAb, MapResidueRing(S, I): AlgEtQOrd, AlgEtQIdl → GrpAb, MapResidueField(P): AlgEtQIdl → FldFin, MapPrimitiveElementResidueField(P): AlgEtQIdl → AlgEtQEltQuotientVS(I, J, P): AlgEtQOrd, AlgEtQOrd, AlgEtQIdl → ModRng, Map- Example: Residue Field and Residue Ring
QuotientVS(I, J, P): AlgEtQOrd, AlgEtQIdl, AlgEtQIdl → ModRng, MapQuotientVS(I, J, P): AlgEtQIdl, AlgEtQOrd, AlgEtQIdl → ModRng, MapQuotientVS(I, J, P): AlgEtQIdl, AlgEtQIdl, AlgEtQIdl → ModRng, Map
- Over Orders
- Over Order Graph
- Trace and Norm
- Completion
- 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
- Minimal Generators
- Chinese Remainder Theorem
- Picard Group
ResidueRingUnits(S, I): AlgEtQOrd, AlgEtQIdl → GrpAb, MapResidueRingUnits(I): AlgEtQIdl → GrpAb, MapResidueRingUnitsSubgroupGenerators(F): AlgEtQIdl → SeqEnum[AlgEtQElt]IsPrincipal(I1): AlgEtQIdl → BoolElt, AlgAssEltPicardGroup(S): AlgEtQOrd → GrpAb, MapExtensionHomPicardGroups(S, T): AlgEtQOrd, AlgEtQOrd → MapUnitGroup(S): AlgEtQOrd → GrpAb, Map- Example: Picard Group and Units of an Order
IsIsomorphic(I, J): AlgEtQIdl, AlgEtQIdl → BoolElt, AlgAssElt
- Factorization and Primes
Factorization(I): AlgEtQIdl → TupPrimesAbove(I): AlgEtQIdl → SeqEnum[AlgAssEtOrdIdl]SingularPrimes(R): AlgEtQOrd → SeqEnum[AlgAssEtOrdIdl]PlacesAboveRationalPrime(E, p): AlgEtQ, RngIntElt → SeqEnum[AlgEtQIdl]NonInvertiblePrimes(R): AlgEtQOrd → SetIndxIsPrime(I): AlgEtQIdl → BoolEltIsMaximal(I): AlgEtQIdl → BoolEltIsMaximalIdeal(I): AlgEtQIdl → BoolEltValuation(x, P): AlgEtQElt, AlgEtQIdl → RngIntEltValuation(I, P): AlgEtQIdl, AlgEtQIdl → RngIntEltInertiaDegree(P): AlgEtQIdl → RngIntEltRamificationIndex(P): AlgEtQIdl → RngIntEltIsBassAtPrime(S, P): AlgEtQOrd, AlgEtQIdl → BoolEltIsBass(S): AlgEtQOrd → BoolEltIsGorensteinAtPrime(S, P): AlgEtQOrd, AlgEtQIdl → BoolEltIsGorenstein(O): AlgEtQOrd → BoolEltUniformizers(PPs): SeqEnum[AlgEtQIdl] → SeqEnum
- Low Cohen Macauley Type
- Weak Classes
WKICM_bar(S): AlgEtQOrd → SeqEnumWeakEquivalenceClassesWithPrescribedMultiplicatorRing(S): AlgEtQOrd → SeqEnum[AlgEtQIdl]WeakEquivalenceClassMonoid(E): AlgEtQOrd → SeqEnum[AlgEtQIdl]WKICM(E): AlgEtQOrd → SeqEnumWeakEquivalenceClassMonoidAbstract(R): AlgEtQOrd → AlgEtQWECM, MapW ! x: AlgEtQWECM, Any → AlgEtQWECMEltx in W: AlgEtQWECMElt, AlgEtQWECM → BoolEltParent(x): AlgEtQWECMElt → AlgEtQWECMIdeal(x): AlgEtQWECMElt → AlgEtQIdlMultiplicatorRing(x): AlgEtQWECMElt → AlgEtQOrdx1 eq x2: AlgEtQWECMElt, AlgEtQWECMElt → BoolEltSetRepresentative(x, J): AlgEtQWECMElt, AlgEtQIdlMultiplicationTable(W): AlgEtQWECM → Assocx * y: AlgEtQWECMElt, AlgEtQWECMElt → AlgEtQWECMEltx ^ n: AlgEtQWECMElt, RngIntElt → AlgEtQWECMEltIsOne(x): AlgEtQWECMElt → BoolEltIsIdempotent(x): AlgEtQWECMElt → BoolEltOrder(W): AlgEtQWECM → AlgEtQOrdArray(W): AlgEtQWECM → AssocRepresentativeMap(W): AlgEtQWECM → MapW1 eq W2: AlgEtQWECM, AlgEtQWECM → BoolElt# W: AlgEtQWECM → RngIntClasses(W): AlgEtQWECM → SeqEnum[AlgEtQWECMElt]Representatives(W): AlgEtQWECM → SeqEnum[AlgEtQIdl]One(W): AlgEtQWECM → AlgEtQWECMEltRandom(W): AlgEtQWECM → AlgEtQWECMEltIdempotents(W): AlgEtQWECM → SeqEnum[AlgEtQWECMElt]Localization(W, P): AlgEtQWECM, AlgEtQIdl → AlgEtQWECMIsGeneratingSet(W, seq): AlgEtQWECM, SeqEnum[AlgEtQWECMElt] → BoolEltIsGeneratingSet(W, seq): AlgEtQWECM, SeqEnum[AlgEtQIdl] → BoolElt- Example: Weak Equivalence Classes
- Weak Testing
IsWeakEquivalent(I, J): AlgEtQIdl, AlgEtQIdl → BoolEltIsWeaklyEquivalent(I, J): AlgEtQIdl, AlgEtQIdl → BoolEltIsWeakEquivalent(O1, O2): AlgEtQOrd, AlgEtQOrd → BoolEltIsWeaklyEquivalent(O1, O2): AlgEtQOrd, AlgEtQOrd → BoolEltIsWeakEquivalent(O, J): AlgEtQOrd, AlgEtQIdl → BoolEltIsWeaklyEquivalent(J, O): AlgEtQIdl, AlgEtQOrd → BoolElt- Example: Testing Weak Equivalence
IsWeakEquivalent(J, O): AlgEtQIdl, AlgEtQOrd → BoolElt
- Ideal Class Monoid
- Complex Conjugation
- Complex Multiplication
- Totally Real and Positive
- Printing and Saving
- Bibliography