Number Fields and Orders
- Introduction
- Acknowledgement
- Creation Functions
- Creation of Algebraic Fields
NumberField(f): RngUPolElt → FldNumRationalsAsNumberField() → FldNumQNF() → FldNumNumberField(s): [ RngUPolElt ] → FldNumext< F | s1, ..., sn >: FldAlg, RngUPolElt, ..., RngUPolElt → FldAlgext<F | s>: FldAlg, [RngUPolElt] → FldAlgext< Q | s1, ..., sn >: FldRat, RngUPolElt, ..., RngUPolElt → FldNumext< Q | s >: FldRat, [RngUPolElt] → FldNumRadicalExtension(F, d, a): Rng, RngIntElt, RngElt → FldAlgSplittingField(F): FldAlg → FldAlg, SeqEnumNormalClosure(F): FldAlg → FldAlg, SeqEnumSplittingField(f): RngUPolElt → FldAlgSplittingField(L): [RngUPolElt] → FldNum, [FldNumElt]sub< F | e₁, ..., eₙ >: FldAlg, FldAlgElt, ..., FldAlgElt → FldAlg, Mapsub< F | S >: FldAlg, SeqEnum → FldAlg, MapMergeFields(F, L): FldAlg, FldAlg → SeqEnumCompositeFields(F, L): FldAlg, FldAlg → SeqEnumCompositum(K, L): FldAlg, FldAlg → FldAlgCompositum(K, A): FldAlg, FldAb → FldAlgOptimizedRepresentation(F): FldAlg → FldAlg, MapOptimisedRepresentation(F): FldAlg → FldAlg, MapOptimizedRepresentation(F, d): FldAlg, RngIntElt → FldAlg, MapOptimisedRepresentation(F, d): FldAlg, RngIntElt → FldAlg, MapExample: Opt Rep Ord
- Creation of Orders and Fields from Orders
EquationOrder(f): RngUPolElt → RngOrdEquationOrder(S): [RngUPolElt] → RngOrdEquationOrder(K): FldNum → RngOrdSubOrder(O): RngOrd → RngOrdEquationOrder(O): RngOrd → RngOrdIntegers(O): RngOrd → RngOrdRingOfIntegers(O): RngOrd → RngOrdIntegerRing(O): RngOrd → RngOrdExample: Orderssub< O | a₁, ..., aᵣ >: RngOrd, RngOrdElt, ..., RngOrdElt → RngOrdext< O | a₁, ..., aᵣ >: RngOrd, RngOrdElt, ..., RngOrdElt → RngOrdext< Z | f >: RngInt, RngUPolElt → RngOrdext< O | f >: RngOrd, RngUPolElt → RngOrdFieldOfFractions(O): RngOrd → FldOrdOrder(F): FldOrd → RngOrdNumberField(O): RngOrd → FldNumNumberField(F): FldOrd → FldNumExample: fractionsOptimizedRepresentation(O): RngOrd → BoolElt, RngOrd, MapOptimisedRepresentation(O): RngOrd → BoolElt, RngOrd, MapOptimizedRepresentation(O, d): RngOrd, RngIntElt → BoolElt, RngOrd, MapOptimisedRepresentation(O, d): RngOrd, RngIntElt → BoolElt, RngOrd, MapO + P: RngOrd, RngOrd → RngOrdO meet P: RngOrd, RngOrd → RngOrdAsExtensionOf(O, P): RngOrd, RngOrd → RngOrdOrder(O, T, d): RngOrd, AlgMatElt, RngIntElt → RngOrdOrder(O, M): RngOrd, ModDed → RngOrdOrder(O, M): RngOrd, PMat → RngOrdOrder( [ e₁, ... eₙ ] ): [FldAlgElt] → RngOrd
- Maximal Orders
- Creation of Elements
F ! a: FldAlg, RngElt → FldAlgEltelt< F | a >: FldAlg, RngElt → FldAlgEltF ! [a₀, a₁, ..., aₘ₋₁]: FldAlg, [RngElt] → FldAlgEltelt< F | [ a₀, a₁, ..., aₘ₋₁ ] >: FldAlg, SeqEnum[RngElt] → FldAlgEltelt< F | a₀, a₁, ..., aₘ₋₁>: FldAlg, RngElt, ..., RngElt → FldAlgEltO ! a: RngOrd, RngElt → RngOrdEltelt< O | a >: RngOrd, RngElt → RngOrdEltO ! [a₀, a₁, ..., aₘ₋₁]: RngOrd, [ RngElt ] → RngOrdEltelt< O | [ a₀, a₁, ..., aₘ₋₁ ] >: RngOrd, SeqEnum[RngElt] → RngOrdEltelt< O | a₀, a₁, ..., aₘ₋₁ >: RngOrd, RngElt, ..., RngElt → RngOrdEltRandom(F, m): FldAlg, RngIntElt → FldAlgEltRandom(O, m): RngOrd, RngIntElt → RngOrdEltRandom(I, m): RngOrdFracIdl, RngIntElt → FldOrdEltExample: ElementsOne(K): FldAlg → FldAlgEltOne(O): RngOrd → RngOrdEltIdentity(K): FldAlg → FldAlgEltIdentity(O): RngOrd → RngOrdEltZero(K): FldAlg → FldAlgEltZero(O): RngOrd → RngOrdEltRepresentative(K): FldAlg → FldAlgEltRepresentative(O): RngOrd → RngOrdElt
- Creation of Homomorphisms
hom< F -> R | r >: FldAlg, Rng, RngElt → Maphom< F -> R | h, r >: FldAlg, Rng, Map, RngElt → Maphom< O -> R | r >: RngOrd, Rng, RngElt → Maphom< O -> R | h, r >: RngOrd, Rng, Map, RngElt → MapExample: Homomorphismshom< O -> R | b₁, ..., bₙ >: RngFunOrd, Rng, RngElt, ..., RngElt → Maphom< O -> R | m, b₁, ..., bₙ >: RngFunOrd, Rng, Map, RngElt, ..., RngElt → MapIsRingHomomorphism(m): Map → BoolElt
- Printing
- Real Precision
SetKantPrecision(F, n): FldAlg, RngIntEltSetKantPrecision(F, n, m): FldAlg, RngIntElt, RngIntEltSetKantPrecision(O, n): RngOrd, RngIntEltSetKantPrecision(O, n, m): RngOrd, RngIntElt, RngIntElt
- Structure Operations
- General Functions
Category(F): FldAlg → CatParent(F): FldAlg → PowCategory(O): RngOrd → CatParent(O): RngOrd → PowAssignNames(~K, s): FldNum, [ MonStgElt ]Name(K, i): FldNum, RngIntElt → FldNumEltK . i: FldNum, RngIntElt → FldNumEltAssignNames(~F, s): FldOrd, [ MonStgElt ]F . i: FldOrd, RngIntElt → FldOrdEltName(F, i): FldOrd, RngIntElt → FldOrdEltO . i: RngOrd, RngIntElt → FldOrdElt
- Related Structures
GroundField(F): FldAlg → FldBaseField(F): FldAlg → FldCoefficientField(F): FldAlg → FldCoefficientRing(F): FldAlg → FldBaseRing(O): RngOrd → RngCoefficientRing(O): RngOrd → RngAbsoluteField(F): FldAlg → FldAlgAbsoluteOrder(O): RngOrd → RngOrdSimpleExtension(F): FldAlg → FldAlgSimpleExtension(O): RngOrd → RngOrdRelativeField(F, L): FldAlg, FldAlg → FldAlgRelativeField(Q, L): FldRat, FldAlg → FldAlgComponents(F): FldAlg → [FldAlg]Components(O): RngOrd → [RngOrd]Example: CompositumSimplify(O): RngOrd → RngOrdLLL(O): RngOrd → RngOrd, AlgMatEltExample: lllPrimeRing(F): FldAlg → RngRatPrimeField(F): FldAlg → RngRatPrimeRing(O): RngOrd → RngIntCentre(F): FldAlg → FldAlgCentre(O): RngOrd → RngOrdEmbed(F, L, a): FldAlg, FldAlg, FldAlgEltEmbed(F, L, a): FldAlg, FldAlg, [FldAlgElt]EmbeddingMap(F, L): FldAlg, FldAlg → MapHasEmbedding(F, L): FldAlg, FldAlg → BoolHasEmbedding(O1, O2): RngOrd, RngOrd → BoolCanComputeEmbedding(F, L): FldAlg, FldAlg → BoolCanComputeEmbedding(O1, O2): RngOrd, RngOrd → BoolExample: emCompletion(K, P): FldAlg, RngOrdIdl → FldLoc, MapCompletion(O, P): RngOrd, RngOrdIdl → RngLoc, Mapcomp<K|P>: FldAlg, RngOrdIdl → FldLoc, Mapcomp<O|P>: RngOrd, RngOrdIdl → RngLoc, MapCompletion(K, P): FldAlg, PlcNumElt → FldLoc, MapLocalRing(P, prec): RngOrdIdl, RngIntElt → RngLoc, MapLocalization(O, P): RngOrd, RngOrdIdl → RngVal, Map
- Representing Fields as Vector Spaces
Algebra(K, J): FldAlg, Fld → AlgAss, MapAlgebra(K, J, S): FldAlg, Fld, [FldAlgElt] → AlgAss, MapVectorSpace(K, J): FldAlg, Fld → ModTupFld, MapKSpace(K, J): FldAlg, Fld → ModTupFld, MapVectorSpace(K, J, S): FldAlg, Fld, [FldAlgElt] → ModTupFld, MapKSpace(K, J, S): FldAlg, Fld, [FldAlgElt] → ModTupFld, MapExample: Vector Space Eg
- Invariants
Characteristic(F): FldAlg → RngIntEltCharacteristic(O): RngOrd → RngIntEltDegree(O): RngOrd → RngIntEltDegree(F): FldAlg → RngIntEltAbsoluteDegree(O): RngOrd → RngIntEltAbsoluteDegree(F): FldAlg → RngIntEltDiscriminant(O): RngOrd → RngIntEltDiscriminant(O): RngInt → RngIntEltDiscriminant(F): FldAlg → RngIntEltDiscriminant(O): RngOrd → RngOrdIdlDiscriminant(F): FldAlg → RngOrdIdlAbsoluteDiscriminant(O): RngOrd → RngIntEltAbsoluteDiscriminant(K): FldAlg → FldRatEltReducedDiscriminant(O): RngOrd → RngIntEltReducedDiscriminant(F): FldAlg → RngIntEltReducedDiscriminant(O): RngOrd → RngOrdIdlReducedDiscriminant(F): FldAlg → RngOrdIdlRegulator(O: parameters): RngOrd → FldReEltRegulator(K): FldNum → FldReEltRegulatorLowerBound(O): RngOrd → FldReEltRegulatorLowerBound(K): FldNum → FldReEltSignature(O): RngOrd → RngIntElt, RngIntEltSignature(F): FldAlg → RngIntElt, RngIntEltUnitRank(O): RngOrd → RngIntEltUnitRank(K): FldNum → RngIntEltIndex(O, S): RngOrd, RngOrd → RngIntEltIndex(O, S): RngOrd, RngOrd → RngOrdIdlDefiningPolynomial(F): FldAlg → RngUPolEltDefiningPolynomial(O): RngOrd → RngUPolEltDefiningPolynomial(F): FldAlg → [RngUPolElt]DefiningPolynomial(O): RngOrd → [RngUPolElt]Zeroes(O, n): RngOrd, RngIntElt → [ FldReElt ]Zeros(O, n): RngOrd, RngIntElt → [ FldReElt ]Zeroes(F, n): FldAlg, RngIntElt → [ FldReElt ]Zeros(F, n): FldAlg, RngIntElt → [ FldReElt ]Example: zeroDifferent(O): RngOrd → RngOrdIdlConductor(O): RngOrd → RngOrdIdl
- Basis Representation
- Ring Predicates
N eq O: RngOrd, RngOrd → BoolEltF eq L: FldAlg, FldAlg → BoolEltF eq L: FldRat, FldAlg → BoolEltF eq L: FldAlg, FldRat → BoolEltIsCommutative(R): Rng → BoolEltIsUnitary(R): Rng → BoolEltIsFinite(R): Rng → BoolEltIsOrdered(R): Rng → BoolEltIsField(R): Rng → BoolEltIsNumberField(R): . → BoolEltIsAlgebraicField(R): Any → BoolEltIsEuclideanDomain(F): FldAlg → BoolEltIsSimple(F): FldAlg → BoolEltIsSimple(O): RngOrd → BoolEltIsPID(F): FldAlg → BoolEltIsUFD(F): FldAlg → BoolEltIsPrincipalIdealRing(F): FldAlg → BoolEltIsPID(O): RngOrd → BoolEltIsUFD(O): RngOrd → BoolEltIsPrincipalIdealRing(O): RngOrd → BoolEltIsDomain(R): FldAlg → BoolEltF ne L: FldAlg, FldAlg → BoolEltO ne N: RngOrd, RngOrd → BoolEltO subset P: RngOrd, RngOrd → BoolEltK subset L: FldAlg, FldAlg → BoolEltK subset L: FldRat, FldAlg → BoolEltHasComplexConjugate(K): FldAlg → BoolElt, MapComplexConjugate(x): FldAlgElt → FldAlgElt
- Order Predicates
- Field Predicates
IsIsomorphic(F, L): FldAlg, FldAlg → BoolElt, MapIsSubfield(F, L): FldAlg, FldAlg → BoolElt, MapIsSubfield(F, L): FldRat, FldAlg → BoolElt, MapIsNormal(F): FldAlg → BoolEltIsAbelian(F): FldAlg → BoolEltIsCyclic(F): FldAlg → BoolEltIsAbsoluteField(K): FldAlg → BoolEltIsWildlyRamified(K): FldAlg → BoolEltIsTamelyRamified(K): FldAlg → BoolEltIsUnramified(K): FldAlg → BoolEltIsQuadratic(K): FldAlg → BoolElt, FldQuadIsTotallyReal(K): FldAlg → BoolElt
- Setting Properties of Orders
- Element Operations
- Parent and Category
- Arithmetic
+ a: FldAlgElt → FldAlgElt+ w: RngOrdElt → RngOrdElt- a: FldAlgElt → FldAlgElt- w: RngOrdElt → RngOrdElta + b: FldAlgElt, FldAlgElt → FldAlgEltw + v: RngOrdElt, RngOrdElt → RngOrdElta - b: FldAlgElt, FldAlgElt → FldAlgEltw - v: RngOrdElt, RngOrdElt → RngOrdElta * b: FldAlgElt, FldAlgElt → FldAlgEltw * v: RngOrdElt, RngOrdElt → RngOrdElta / b: FldAlgElt, FldAlgElt → FldAlgEltw / v: RngOrdElt, RngOrdElt → FldOrdElta ^ k: FldAlgElt, RngIntElt → FldAlgEltw ^ k: RngOrdElt, RngIntElt → RngOrdEltw div v: RngOrdElt, RngOrdElt → RngOrdEltModexp(a, n, m): RngOrdElt, RngIntElt, RngIntElt → RngOrdEltSqrt(a): RngOrdElt → RngOrdEltSqrt(a): FldAlgElt → FldAlgEltSquareRoot(a): RngOrdElt → RngOrdEltSquareRoot(a): FldAlgElt → FldAlgEltRoot(a, n): RngOrdElt, RngIntElt → RngOrdEltRoot(a, n): FldAlgElt, RngIntElt → FldAlgEltIsPower(a, k): FldAlgElt, RngIntElt → BoolElt, FldAlgEltIsPower(a, k): RngOrdElt, RngIntElt → BoolElt, RngOrdEltIsSquare(a): FldAlgElt → BoolElt, FldAlgEltIsSquare(a): RngOrdElt → BoolElt, RngOrdEltDenominator(a): FldAlgElt → RngIntEltDenominator(a): RngOrdElt → RngIntEltNumerator(a): FldAlgElt → RngIntEltQround(E, M): FldAlgElt, RngIntElt → FldAlgElt
- Equality and Membership
- Predicates on Elements
- Field Generators
- Real and Complex Embeddings
Conjugates(a): FldAlgElt → [ FldComElt ]Conjugates(a): RngOrdElt → [ FldComElt ]Conjugate(a, l): FldAlgElt, RngIntElt → FldReEltConjugate(a, l): RngOrdElt, RngIntElt → FldReEltConjugate(a, l): FldAlgElt, [RngIntElt] → FldReEltAbsoluteValues(a): FldAlgElt → [FldReElt]AbsoluteValues(a): RngOrdElt → [FldReElt]Logs(a): FldAlgElt → [FldReElt]Logs(a): RngOrdElt → [FldReElt]InfinitePlaces(K): FldAlg → [PlcNumElt]InfinitePlaces(O): RngOrd → [PlcNumElt]Evaluate(x, p): FldAlgElt, PlcNumElt → RngEltEvaluate(x, p): RngOrdElt, PlcNumElt → RngEltRealEmbeddings(a): FldAlgElt → []RealEmbeddings(a): RngOrdElt → []MinkowskiLattice(O): RngOrd → Lat, MapLattice(O): RngOrd → Lat, MapMinkowskiLattice(I): RngOrdIdl → Lat, MapLattice(I): RngOrdIdl → Lat, MapMinkowskiSpace(F): FldAlg → Lat, Map
- Heights
- Norm, Trace, and Minimal Polynomial
Norm(a): FldAlgElt → FldAlgEltNorm(a): FldAlgElt → FldRatEltNorm(a, R): FldAlgElt, Rng → RngEltNorm(a): RngOrdElt → RngOrdEltNorm(a): RngOrdElt → RngIntEltNorm(a, R): RngOrdElt, Rng → RngEltAbsoluteNorm(a): FldAlgElt → FldRatEltNormAbs(a): FldAlgElt → FldRatEltAbsoluteNorm(a): RngOrdElt → FldRatEltNormAbs(a): RngOrdElt → FldRatEltTrace(a): FldAlgElt → FldAlgEltTrace(a): FldAlgElt → FldRatEltTrace(a, R): FldAlgElt, Rng → RngEltTrace(a): RngOrdElt → RngOrdEltTrace(a): RngOrdElt → RngIntEltTrace(a, R): RngOrdElt, Rng → RngEltAbsoluteTrace(a): FldAlgElt → FldRatEltTraceAbs(a): FldAlgElt → FldRatEltAbsoluteTrace(a): RngOrdElt → FldRatEltTraceAbs(a): RngOrdElt → FldRatEltCharacteristicPolynomial(a): FldAlgElt → RngUPolEltCharacteristicPolynomial(a, R): FldAlgElt, Rng → RngUPolEltCharacteristicPolynomial(a): RngOrdElt → RngUPolEltCharacteristicPolynomial(a, R): RngOrdElt, Rng → RngUPolEltAbsoluteCharacteristicPolynomial(a): FldAlgElt → RngUPolEltAbsoluteCharacteristicPolynomial(a): RngOrdElt → RngUPolEltMinimalPolynomial(a): FldAlgElt → RngUPolEltMinimalPolynomial(a, R): FldAlgElt, Rng → RngUPolEltMinimalPolynomial(a): RngOrdElt → RngUPolEltMinimalPolynomial(a, R): RngOrdElt, Rng → RngUPolEltAbsoluteMinimalPolynomial(a): FldAlgElt → RngUPolEltAbsoluteMinimalPolynomial(a): RngOrdElt → RngUPolEltRepresentationMatrix(a): FldAlgElt → AlgMatEltRepresentationMatrix(a, R): FldAlgElt, Rng → AlgMatEltRepresentationMatrix(a): RngOrdElt → AlgMatEltRepresentationMatrix(a, R): RngOrdElt, Rng → AlgMatEltAbsoluteRepresentationMatrix(a): FldAlgElt → AlgMatEltAbsoluteRepresentationMatrix(a): RngOrdElt → AlgMatEltExample: Norms Etc
- The Quadratic Defect
QuadraticDefect(a, p): RngElt, RngOrdIdl → RngIntEltQuadraticDefect(a, p): FldRatElt, RngIntElt → RngIntEltQuadraticDefect(a, p): RngIntElt, RngIntElt → RngIntEltQuadraticDefect(a, p): FldRatElt, RngInt → RngIntEltQuadraticDefect(a, p): RngIntElt, RngInt → RngIntEltRelativeQuadraticDefect(a, p): RngElt, RngOrdIdl → RngIntEltRelativeQuadraticDefect(a, p): FldRatElt, RngIntElt → RngIntEltRelativeQuadraticDefect(a, p): RngIntElt, RngIntElt → RngIntEltRelativeQuadraticDefect(a, p): FldRatElt, RngInt → RngIntEltRelativeQuadraticDefect(a, p): RngIntElt, RngInt → RngIntEltIsLocalSquare(a, p): RngElt, RngOrdIdl → BoolEltIsLocalSquare(a, p): FldRatElt, RngIntElt → BoolEltIsLocalSquare(a, p): RngIntElt, RngIntElt → BoolEltIsLocalSquare(a, p): FldRatElt, RngInt → BoolEltIsLocalSquare(a, p): RngIntElt, RngInt → BoolEltLocalMultiplicativeGroupModSquares(p): RngOrdIdl → ModFld, MapLocalMultiplicativeGroupModSquares(p): RngIntElt → ModFld, MapUnitSquareClassReps(p): RngOrdIdl → SeqEnumUnitSquareClassReps(p): RngIntElt → SeqEnumNiceUnitSquareClassRepresentative(u, p): RngElt, RngOrdIdl → RngEltNiceUnitSquareClassRepresentative(u, p): RngElt, RngIntElt → RngElt
- Other Functions
ElementToSequence(a): FldAlgElt → [ FldAlgElt ]Eltseq(a): FldAlgElt → [ FldAlgElt ]ElementToSequence(a): RngOrdElt → [ FldOrdElt ]Eltseq(a): RngOrdElt → [ FldOrdElt ]Eltseq(E, k): FldAlgElt, FldAlg → [RngElt]Eltseq(E, k): FldAlgElt, Rng → [RngElt]Flat(e): FldAlgElt → [FldRatElt]a[i]: FldAlgElt, RngIntElt → FldRatElta[i]: FldAlgElt, RngIntElt → FldAlgElta[i]: RngOrdElt, RngIntElt → FldRatElta[i]: RngOrdElt, RngIntElt → FldOrdEltProductRepresentation(a): RngOrdElt → [ RngOrdElt ], [ RngIntElt ]ProductRepresentation(a): FldAlgElt → [ FldAlgElt ], [ RngIntElt ]ProductRepresentation(P, E): [ FldAlgElt ], [ RngIntElt ] → FldAlgEltPowerProduct(P, E): [FldAlgElt], [RngIntElt] → FldAlgEltValuation(w, I): RngOrdElt, RngOrdIdl → RngIntEltValuation(w, I): FldAlgElt, RngOrdIdl → RngIntEltDecomposition(a): RngOrdElt → SeqEnum[<RngOrdIdl, RngIntElt>]Decomposition(a): FldOrdElt → SeqEnum[<RngOrdIdl, RngIntElt>]Divisors(a): RngOrdElt → SeqEnum[RngOrdElt]Index(a): RngOrdElt → RngIntEltDifferent(a): RngOrdElt → RngOrdEltDegreeOnePrimeIdeals(O, B): RngOrd, RngIntElt → [ RngOrdIdl ]
- Ideal Class Groups
SetPrintClassGroupWarnings(b): BoolEltClassGroup(O: parameters): RngOrd → GrpAb, MapClassGroup(K: parameters): FldNum → GrpAb, MapRingClassGroup(O): RngOrd → GrpAb, MapPicardGroup(O): RngOrd → GrpAb, MapConditionalClassGroup(O): RngOrd → GrpAb, MapConditionalClassGroup(K): FldNum → GrpAb, MapClassGroupPrimeRepresentatives(O, I): RngOrd, RngOrdIdl → MapClassNumber(O: parameters): RngOrd → RngIntEltClassNumber(K: parameters): FldNum → RngIntEltMinkowskiBound(K): FldNum → RngIntEltMinkowskiBound(O): RngOrd → RngIntEltBachBound(K): FldNum → RngIntEltBachBound(O): RngOrd → RngIntEltGRHBound(K): FldNum → RngIntEltGRHBound(O): RngOrd → RngIntEltFactorBasisVerify(O, a, b): RngOrd, RngIntElt, RngIntElt- Class Group Internals
EulerProduct(O, B): RngOrd, RngIntElt → FldReEltResidueGRH(O, B): RngOrd, RngIntElt → FldReEltResidueGRHbound(O, e): RngOrd, FldReElt → FldReEltFactorBasis(K, B): FldNum, RngIntElt → [ RngOrdIdl ]FactorBasis(O, B): RngOrd, RngIntElt → [ RngOrdIdl ]FactorBasis(O): RngOrd → [ RngOrdIdl ], IntegerRelationMatrix(O): RngOrd → ModHomEltRelations(O): RngOrd → ModHomEltClassGroupCyclicFactorGenerators(O): RngOrd → ModHomEltExample: Class Group
- Setting the Class Group Bounds
- Class Group Map Caching
- Unit Groups
UnitRank(O): RngOrd → RngIntEltUnitRank(K): FldAlg → RngIntEltTorsionUnitGroup(O): RngOrd → GrpAb, MapTorsionUnitGroup(K): FldNum → GrpAb, MapUnitGroup(O): RngOrd → GrpAb, MapMultiplicativeGroup(O): RngOrd → GrpAb, MapUnitGroup(K): FldNum → GrpAb, MapMultiplicativeGroup(K): FldNum → GrpAb, MapIndependentUnits(O): RngOrd → GrpAb, MapIndependentUnits(K): FldNum → GrpAb, MappFundamentalUnits(O, p): RngOrd, RngIntElt → GrpAb, MappFundamentalUnits(K, p): FldNum, RngIntElt → GrpAb, MapUnitGroupAsSubgroup(O): RngOrd → GrpAbMergeUnits(K, a): FldNum, FldNumElt → BoolEltMergeUnits(O, a): RngOrd, RngOrdElt → BoolEltExample: Unit GroupIsExceptionalUnit(u): RngOrdElt → BoolEltExceptionalUnitOrbit(u): RngOrdElt → [ RngOrdElt ]ExceptionalUnits(O): RngOrd → [ RngOrdElt ]UnitsWithSigns(O, oo, Signs): RngOrd, [ PlcNumElt ], [ RngInt ] → [ RngOrdElt ]UnitsWithSigns(K, oo, Signs): FldAlg, [ PlcNumElt ], [ RngInt ] → [ FldAlgElt ]UnitsWithSigns(O, x): RngOrd, RngElt → [ RngOrdElt ]UnitsWithSigns(K, x): FldAlg, RngElt → [ FldAlgElt ]UnitsWithSigns(x): RngOrdElt → [ RngOrdElt ]UnitsWithSigns(x): FldAlgElt → [ FldAlgElt ]HasTotallyPositiveGenerator(I): RngOrdFracIdl → BoolElt, [ RngOrdElt ]TraceMinimalCone(nf): FldNum → TorCon
- Diophantine Equations
- Norm Equations
NormEquation(O, m): RngOrd, RngIntElt → BoolElt, [ RngOrdElt ]NormEquation(O, m): RngOrd, RngOrdElt → BoolElt, [ RngOrdElt ]NormEquation(F, m): FldAlg, RngIntElt → BoolElt, [ FldAlgElt ]NormEquation(F, m): FldAlg, FldAlgElt → BoolElt, [ FldAlgElt ]NormEquation(F, m): FldAlg, FldRatElt → BoolElt, [ FldAlgElt ]NormEquation(m, N): RngElt, Map → BoolElt, RngEltIntegralNormEquation(a, N, O): RngElt, Map, RngOrd → BoolElt, [RngOrdElt]SimNEQ(K, e, f): FldNum, FldNumElt, FldNumElt → BoolElt, [FldNumElt]Example: Norm Equation
- Thue Equations
- Unit Equations
- Index Form Equations
- Ideals and Quotients
- Creation of Ideals in Orders
x * O: RngElt, RngOrd → RngOrdFracIdlO * x: RngElt, RngOrd → RngOrdFracIdlF !! I: RngOrd, RngInt → RngOrdFracIdlF !! I: FldOrd, RngOrdFracIdl → RngOrdFracIdlF !! I: FldRat, RngOrdFracIdl → RngIntFracIdlF !! I: FldRat, RngIntFracIdl → RngIntFracIdlF !! I: FldOrd, RngIntFracIdl → RngOrdFracIdlO !! I: RngOrd, RngOrdFracIdl → RngOrdIdlO !! I: RngInt, RngInt → RngIntideal< O | a₁, a₂, ... , aₘ >: RngOrd, RngElt, ..., RngElt → RngOrdFracIdlideal<O | x>: RngOrd, RngIntElt → RngOrdIdlideal<O | M, d>: RngOrd, AlgMatElt, RngIntElt → RngOrdFracIdlideal<O | M, d>: RngOrd, ModDed, RngIntElt → RngOrdFracIdlideal<O | M, I1, ..., In>: RngOrd, AlgMatElt, RngOrdFracIdl, ..., RngOrdFracIdl → RngOrdFracIdlideal<O | M, [I1, ..., In]>: RngOrd, AlgMatElt, [RngOrdFracIdl] → RngOrdFracIdlExample: IdealsFractionalIdeal(x): FldRatElt → RngIntFracIdlFractionalIdeal(x): RngIntElt → RngIntFracIdlFractionalIdeal(x): [ FldRatElt ] → RngIntFracIdlFractionalIdeal(x): [ RngIntElt ] → RngIntFracIdlFractionalIdeal(I): RngInt → RngIntFracIdl
- Invariants
Order(I): RngOrdFracIdl → RngOrdOrder(I): RngIntFracIdl → RngIntDenominator(I): RngOrdFracIdl → RngIntEltDenominator(I): RngIntFracIdl → RngIntEltPrimitiveElement(I): RngOrdIdl → RngOrdEltPrimitiveElement(I): RngOrdFracIdl → FldOrdEltUniformizingElement(P): RngOrdIdl → RngOrdEltUniformizingElement(P): RngInt → RngIntEltIndex(O, I): RngOrd, RngOrdIdl → RngIntEltNorm(I): RngOrdIdl → RngIntEltNorm(I): RngOrdFracIdl → FldRatEltNorm(I): RngOrdFracIdl → RngOrdFracIdlNorm(I): RngIntFracIdl → RngIntEltMinimalInteger(I): RngOrdIdl → RngEltMinimum(I): RngOrdFracIdl → RngEltMin(I): RngOrdFracIdl → RngEltAbsoluteNorm(I): RngOrdIdl → RngIntEltAbsoluteNorm(I): RngOrdFracIdl → FldRatEltNormAbs(I): RngOrdIdl → RngIntEltNormAbs(I): RngOrdFracIdl → FldRatEltCoefficientHeight(I): RngOrdIdl → RngIntEltCoefficientHeight(I): RngOrdFracIdl → RngIntEltCoefficientLength(I): RngOrdIdl → RngIntEltCoefficientLength(I): RngOrdFracIdl → RngIntEltRamificationIndex(I, p): RngOrdIdl, RngIntElt → RngIntEltRamificationDegree(I, p): RngOrdIdl, RngIntElt → RngIntEltRamificationDegree(I): RngOrdIdl → RngIntEltRamificationIndex(I): RngOrdIdl → RngIntEltAbsoluteRamificationDegree(I): RngOrdIdl → RngIntEltAbsoluteRamificationIndex(I): RndOrdIdl → RngIntEltResidueClassField(O, I): RngOrd, RngOrdIdl → FldFin, MapResidueClassField(I): RngOrdIdl → FldFin, MapDegree(I): RngOrdIdl → RngIntEltInertiaDegree(I): RngOrdIdl → RngIntEltAbsoluteInertiaDegree(I): RngOrdIdl → RngIntEltAbsoluteInertiaIndex(I): RngOrdIdl → RngIntEltValuation(I, p): RngOrdFracIdl, RngOrdIdl → RngIntEltValuation(I, p): RngIntFracIdl, RngInt → RngIntEltContent(I): RngOrdFracIdl → RngIntEltExample: Ideal Invar
- Basis Representation
- Two–Element Presentations
- Standard Names
- Predicates on Ideals
I eq J: RngOrdFracIdl, RngOrdFracIdl → BoolEltI ne J: RngOrdFracIdl, RngOrdFracIdl → BoolEltx in I: FldOrdElt, RngOrdFracIdl → BoolEltx in I: RngOrdElt, RngOrdFracIdl → BoolEltx notin I: FldOrdElt, RngOrdFracIdl → BoolEltx notin I: RngOrdElt, RngOrdFracIdl → BoolEltI subset J: RngOrdIdl, RngOrdIdl → BoolEltIsIntegral(I): RngOrdFracIdl → BoolEltIsIntegral(I): RngIntFracIdl → BoolEltIsZero(I): RngOrdFracIdl → BoolEltIsOne(I): RngOrdIdl → BoolEltIsOne(I): RngOrdFracIdl → BoolEltIsOne(I): RngIntFracIdl → BoolEltIsPrime(I): RngOrdIdl → BoolElt, RngOrdIdlIsPrime(I): RngOrdFracIdl → BoolEltIsPrincipal(I): RngOrdFracIdl → BoolElt, FldOrdEltIsRamified(P): RngOrdIdl → BoolEltIsRamified(P, O): RngOrdIdl, RngOrd → BoolEltIsRamified(P, O): RngIntElt, RngOrd → BoolEltIsRamified(P, O): RngInt, RngOrd → BoolEltIsSquarefree(I): RngOrdIdl → BoolEltIsSquarefree(I): RngInt → BoolEltIsTotallyRamified(P): RngOrdIdl → BoolEltIsTotallyRamified(P, O): RngOrdIdl, RngOrd → BoolEltIsTotallyRamified(P, O): RngIntElt, RngOrd → BoolEltIsTotallyRamified(K): FldAlg → BoolEltIsTotallyRamified(O): RngOrd → BoolEltIsWildlyRamified(P): RngOrdIdl → BoolEltIsWildlyRamified(P, O): RngOrdIdl, RngOrd → BoolEltIsWildlyRamified(P, O): RngIntElt, RngOrd → BoolEltIsTamelyRamified(P): RngOrdIdl → BoolEltIsTamelyRamified(P, O): RngOrdIdl, RngOrd → BoolEltIsTamelyRamified(P, O): RngIntElt, RngOrd → BoolEltIsUnramified(P): RngOrdIdl → BoolEltIsUnramified(P, O): RngOrdIdl, RngOrd → BoolEltIsUnramified(P, O): RngIntElt, RngOrd → BoolEltIsInert(P): RngOrdIdl → BoolEltIsInert(P, O): RngOrdIdl, RngOrd → BoolEltIsInert(P, O): RngIntElt, RngOrd → BoolEltIsInert(P, O): RngInt, RngOrd → BoolEltIsSplit(P): RngOrdIdl → BoolEltIsSplit(P, O): RngOrdIdl, RngOrd → BoolEltIsSplit(P, O): RngIntElt, RngOrd → BoolEltIsSplit(P, O): RngInt, RngOrd → BoolEltIsTotallySplit(P): RngOrdIdl → BoolEltIsTotallySplit(P, O): RngOrdIdl, RngOrd → BoolEltIsTotallySplit(P, O): RngIntElt, RngOrd → BoolEltIsTotallySplit(P, O): RngInt, RngOrd → BoolElt
- Ideal Arithmetic
I * J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlI * J: RngIntFracIdl, RngOrdFracIdl → RngOrdFracIdlI * J: RngIntFracIdl, RngIntFracIdl → RngIntFracIdlx * I: RngElt, RngOrdFracIdl → RngOrdFracIdlx * I: FldRatElt, RngIntFracIdl → RngIntFracIdlx * I: RngIntElt, RngIntFracIdl → RngIntFracIdlI * x: RngOrdFracIdl, RngElt → RngOrdFracIdl&* L: [RngOrdFracIdl] → RngOrdFracIdl&* L: { RngOrdFracIdl } → RngOrdFracIdlI div J: RngOrdIdl, RngOrdIdl → RngOrdIdlI div J: RngOrdFracIdl, RngOrdFracIdl → RngOrdIdlI div J: RngInt, RngInt → RngIntI / J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlI / x: RngOrdFracIdl, RngElt → RngOrdFracIdlI + J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlI + J: RngIntFracIdl, RngIntFracIdl → RngIntFracIdlI + J: RngIntFracIdl, RngInt → RngIntFracIdlI ^ k: RngOrdFracIdl, RngIntElt → RngOrdFracIdlI ^ k: RngIntFracIdl, RngIntElt → RngIntFracIdlI eq J: RngOrdFracIdl, RngOrdFracIdl → BoolEltI eq J: RngIntFracIdl, RngIntFracIdl → BoolEltI eq J: RngIntFracIdl, RngInt → BoolEltI subset J: RngOrdIdl, RngOrdIdl → BoolEltE in I: RngOrdElt, RngOrdIdl → BoolEltE in I: RngOrdElt, RngOrdFracIdl → BoolEltE in I: FldAlgElt, RngOrdFracIdl → BoolEltLCM(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlLcm(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlLeastCommonMultiple(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlGCD(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlGcd(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlGreatestCommonDivisor(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlContent(M): Mtrx → RngOrdFracIdlI meet J: RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdl&meet S: [RngOrdFracIdl] → RngOrdFracIdl&meet S: { RngOrdFracIdl } → RngOrdFracIdlI meet R: RngOrdFracIdl, Rng → AnyR meet I: Rng, RngOrdFracIdl → Anya mod I: RngOrdElt, RngOrdIdl → RngOrdEltInverseMod(E, M): RngOrdElt, RngIntElt → RngOrdEltInverseMod(E, M): RngOrdElt, RngOrdIdl → RngOrdEltModinv(E, M): RngOrdElt, RngOrdIdl → RngOrdEltModinv(E, M): RngOrdElt, RngIntElt → RngOrdEltColonIdeal(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlIdealQuotient(I, J): RngOrdFracIdl, RngOrdFracIdl → RngOrdFracIdlExample: colonIntegralSplit(I): RngOrdFracIdl → RngOrdIdl, RngEltDifferent(I): RngOrdFracIdl → RngOrdFracIdlCodifferent(I): RngOrdFracIdl → RngOrdFracIdl
- Roots of Ideals
- Factorization and Primes
Decomposition(O, p): RngOrd, RngIntElt → [<RngOrdIdl, RngIntElt>]Decomposition(O, p): RngOrd, RngOrdIdl → [ <RngOrdIdl, RngIntElt> ]DecompositionType(O, p): RngOrd, RngIntElt → [<RngIntElt, RngIntElt>]DecompositionType(O, p): FldAlg, RngIntElt → [<RngIntElt, RngIntElt>]DecompositionType(O, p): FldAlg, RngOrdIdl → [<RngIntElt, RngIntElt>]DecompositionType(O, p): RngOrd, RngOrdIdl → [ <RngIntElt, RngIntElt> ]Factorization(I): RngOrdFracIdl → [<RngOrdIdl, RngIntElt>]Factorisation(I): RngOrdFracIdl → [<RngOrdIdl, RngIntElt>]Factorization(I): RngIntFracIdl → [<RngInt, RngIntElt>]Factorisation(I): RngIntFracIdl → [<RngInt, RngIntElt>]Example: Non Maximal FactDivisors(I): RngOrdIdl → [<RngOrdIdl, RngIntElt>]Support(I): RngOrdFracIdl → {RngOrdIdl}Support(I): RngIntFracIdl → {RngInt}Support(L): [RngOrdFracIdl] → {RngOrdIdl}Support(L): [FldAlgElt] → {RngOrdIdl}CoprimeBasis(L): [RngOrdFracIdl] → RngOrdIdlCoprimeBasisInsert(~L, I): [RngOrdIdl], RngOrdFracIdlPowerProduct(B, E): [RngOrdFracIdl], [RngIntElt] → RngOrdFracIdl
- Other Ideal Operations
ChineseRemainderTheorem(I1, I2, e1, e2): RngOrdIdl, RngOrdIdl, RngOrdElt, RngOrdElt → RngOrdEltChineseRemainderTheorem(X, M): SeqEnum[RngOrdElt], SeqEnum[RngOrdIdl] → RngOrdEltCRT(I1, I2, e1, e2): RngOrdIdl, RngOrdIdl, RngOrdElt, RngOrdElt → RngOrdEltCRT(X, M): SeqEnum[RngOrdElt], SeqEnum[RngOrdIdl] → RngOrdEltCRT(I1, L1, e1, L2): RngOrdIdl, [RngIntElt], RngOrdElt, [RngIntElt] → RngOrdEltChineseRemainderTheorem(I1, L1, e1, L2): RngOrdIdl, [RngIntElt], RngOrdElt, [RngIntElt] → RngOrdEltWeakApproximation(I, V): [RngOrdIdl], [RngIntElt] → FldOrdEltWeakApproximation(I, V): [RngInt], [RngIntElt] → FldRatEltIdempotents(I, J): RngOrdIdl, RngOrdIdl → BoolElt, RngOrdElt, RngOrdEltCoprimeRepresentative(I, J): RngOrdIdl, RngOrdIdl → FldOrdEltMakeCoprime(I, J): RngOrdIdl, RngOrdIdl → FldOrdEltIdealsUpTo(B, O): RngIntElt, RngOrd → [RngOrdIdl]IdealsUpTo(B, K): RngIntElt, FldNum → [RngOrdIdl]ClassRepresentative(I): RngOrdFracIdl → RngOrdFracIdlSUnitGroup(I): RngOrdFracIdl → GrpAb, MapSUnitGroup(S): [ RngOrdIdl ] → GrpAb, MapExample: S UnitsSUnitAction(SU, Act, S): Map, Map, SeqEnum[RngOrdIdl] → MapSUnitAction(SU, Act, S): Map, SeqEnum[Map], SeqEnum[RngOrdIdl] → [Map]SUnitDiscLog(SU, x, S): Map, FldAlgElt, SeqEnum[RngOrdIdl] → GrpAbEltSUnitDiscLog(SU, L, S): Map, SeqEnum[FldAlgElt], SeqEnum[RngOrdIdl] → GrpAbEltExample: S-Units, advanced
- Quotient Rings
- Operations on Quotient Rings
- Elements of Quotients
OQ ! a: RngOrdRes, Elt → RngOrdResEltRandom(OQ): RngOrdRes → RngOrdResElta mod I: RngOrdElt, RngOrdIdl → RngOrdElta * b: RngOrdResElt, RngOrdResElt → RngOrdResElta + b: RngOrdResElt, RngOrdResElt → RngOrdResElta - b: RngOrdResElt, RngOrdResElt → RngOrdResElta / b: RngOrdResElt, RngOrdResElt → RngOrdResElt- a: RngOrdResElt → RngOrdResElta ^ n: RngOrdResElt, RngIntElt → RngOrdResElta eq b: RngOrdResElt, RngOrdResElt → BoolElta ne b: RngOrdResElt, RngOrdResElt → BoolElta div b: RngOrdResElt, RngOrdResElt → RngOrdResElta mod b: RngOrdResElt, RngOrdResElt → RngOrdResEltQuotrem(a, b): RngOrdResElt, RngOrdResElt → RngOrdResElt, RngOrdResEltLcm(a, b): RngOrdResElt, RngOrdResElt → RngOrdResEltGcd(a, b): RngOrdResElt, RngOrdResElt → RngOrdResEltXGcd(a, b): RngOrdResElt, RngOrdResElt → RngOrdResElt, RngOrdResElt, RngOrdResEltIsZero(a): RngOrdResElt → BoolEltIsOne(a): RngOrdResElt → BoolEltIsMinusOne(a): RngOrdResElt → BoolEltIsUnit(a): RngOrdResElt → BoolEltEltseq(a): RngOrdResElt → []ElementToSequence(a): RngOrdResElt → []EuclideanNorm(a): RngOrdResElt → RngIntEltAnnihilator(a): RngOrdResElt → RngOrdResElt
- Reconstruction
ReconstructionEnvironment(p, k): RngOrdIdl, RngIntElt → RngOrdRecoEnvReconstructionEnvironment(p, k): RngIntElt, RngIntElt → RngOrdRecoEnvReconstruct(x, R): RngOrdElt, RngOrdRecoEnv → RngOrdEltReconstruct(x, R): RngIntElt, RngOrdRecoEnv → RngIntEltChangePrecision(~R, k): RngOrdRecoEnv, RngIntEltExample: Order Reco
- Places and Divisors
- Creation of Structures
- Operations on Structures
- Creation of Elements
Place(I): RngOrdIdl → PlcNumEltDecomposition(K, I): FldAlg, Infty → SeqEnumDecomposition(K, p): FldAlg, RngIntElt → SeqEnumDecomposition(K, p): FldNum, PlcNumElt → SeqEnumDecomposition(m, p): Map[FldRat, FldAlg], RngIntElt → SeqEnum[<PlcNumElt, RngIntElt>]Decomposition(m, p): Map[FldAlg, FldAlg], PlcNumElt → SeqEnum[<PlcNumElt, RngIntElt>]InfinitePlaces(K): FldAlg → [PlcNumElt]InfinitePlaces(O): RngOrd → [PlcNumElt]RealPlaces(K): FldAlg → [PlcNumElt]Divisor(pl): PlcNumElt → DivNumEltDivisor(I): RngOrdFracIdl → DivNumEltDivisor(x): FldNumElt → DivNumElt
- Arithmetic with Places and Divisors
d1 + d2: DivNumElt, DivNumElt → DivNumEltp + d: PlcNumElt, DivNumElt → DivNumEltd + p: DivNumElt, PlcNumElt → DivNumEltp1 + p2: PlcNumElt, PlcNumElt → DivNumElt- p: PlcNumElt → DivNumElt- d: DivNumElt → DivNumEltd - p: DivNumElt, PlcNumElt → DivNumEltp - d: PlcNumElt, DivNumElt → DivNumEltd1 - d2: DivNumElt, DivNumElt → DivNumEltp1 - p2: PlcNumElt, PlcNumElt → DivNumEltp * k: PlcNumElt, RngIntElt → DivNumEltd * k: DivNumElt, RngIntElt → DivNumEltp div k: PlcNumElt, RngIntElt → DivNumEltd div k: DivNumElt, RngIntElt → DivNumElt
- Other Functions for Places and Divisors
Valuation(a, p): FldNumElt, PlcNumElt → RngEltValuation(a, p): RngElt, PlcNumElt → RngEltValuation(I, p): RngOrdFracIdl, PlcNumElt → RngEltSupport(D): DivNumElt → SeqEnum, SeqEnumIdeal(D): DivNumElt → RngOrdIdlIdeal(D): PlcNumElt → RngOrdIdlIsFinite(p): PlcNumElt → BoolEltIsInfinite(p): PlcNumElt → BoolElt, RngIntEltIsReal(p): PlcNumElt → BoolEltIsComplex(p): PlcNumElt → BoolEltExtends(P, p): PlcNumElt, PlcNumElt → BoolEltInertiaDegree(P): PlcNumElt → RngIntEltDegree(P): PlcNumElt → RngIntEltDegree(D): DivNumElt → RngEltNumberField(P): PlcNumElt → FldNumNumberField(D): DivNumElt → FldNumResidueClassField(P): PlcNumElt → FldUniformizingElement(P): PlcNumElt → FldNumEltLocalDegree(P): PlcNumElt → RngIntEltRamificationIndex(P): PlcNumElt → RngIntEltDecompositionGroup(P): PlcNumElt → GrpPerm
- The Montes Algorithm
Montes(f, p): RngUPolElt, RngElt → SeqEnum, SeqEnum, RngIntEltExample: Montes Eg 1Montes(K, p): FldArith, RngEltExample: Montes Eg 2SFL(P, s): OMIdl, RngIntEltExample: sflSetUseMontes(f): BoolEltSetUseMontes(t, f): Cat, BoolEltGetUseMontes(t): Cat → BoolEltSetVerbose("Montes", v): MonStgElt, RngIntElt
- Ideals in OM Representation
- Ideal Operations
pIntegralBasis(I, p): OMIdl, RngElt → SeqEnumSIntegralBasis(I, S): OMIdl, SeqEnum → SeqEnumSIntegralBasis(I, S): RngOrdFracIdl, [RngIntElt] → SeqEnumSIntegralBasis(I, S): RngFunOrdIdl, [RngUPolElt] → SeqEnumBasis(I): OMIdl → SeqEnumExample: Om Ideal OpTwoElement(I): OMIdl → FldArithElt, FldArithEltNorm(I): OMIdl → RngEltValuation(alpha, P : parameters): FldArithElt, OMIdl → RngIntElt, FldEltValuation(alpha, P : parameters): FldRatElt, OMIdl → RngIntElt, FldEltValuation(alpha, P : parameters): RngIntElt, OMIdl → RngIntElt, FldEltValuation(alpha, P : parameters): RngUPolElt, OMIdl → RngIntElt, FldEltValuation(I, P): OMIdl, OMIdl → RngIntElta mod P: FldArithElt, OMIdl → FldArithEltReduction(a, P): FldArithElt, OMIdl → FldArithEltReduction(a, P, m): FldArithElt, OMIdl, RngIntElt → [FldArithElt]Factorization(I): OMIdl → SeqEnumFactorisation(I): OMIdl → SeqEnumExample: Om Ideal OpsResidueField(I): OMIdl → FldExample: Om Ideals Deg Res
