Number Fields
- Introduction
- Acknowledgement
- Creation Functions
- Creation of Number Fields
NumberField(f): RngUPolElt → FldNumRationalsAsNumberField() → FldNumQNF() → FldNumNumberField(s): [ RngUPolElt ] → FldNumext< F | s1, ..., sn >: FldNum, RngUPolElt, ..., RngUPolElt → FldNumext<F | s>: FldNum, [RngUPolElt] → FldNumext< Q | s1, ..., sn >: FldRat, RngUPolElt, ..., RngUPolElt → FldNumext< Q | s >: FldRat, [RngUPolElt] → FldNumExample: CreationRadicalExtension(F, d, a): Rng, RngIntElt, RngElt → FldNumSplittingField(F): FldNum → FldNum, SeqEnumNormalClosure(F): FldNum → FldNum, SeqEnumSplittingField(f): RngUPolElt → FldNumSplittingField(L): [RngUPolElt] → FldNum, [FldNumElt]sub< F | e₁, ..., eₙ >: FldAlg, FldAlgElt, ..., FldAlgElt → FldAlg, Mapsub< F | S >: FldAlg, SeqEnum → FldAlg, MapMergeFields(F, L): FldNum, FldNum → SeqEnumCompositeFields(F, L): FldNum, FldNum → SeqEnumCompositum(K, L): FldNum, FldNum → FldNumquo< FldNum : R | f >: RngUPol, RngUPolElt → FldNumExample: Composite FieldsOptimizedRepresentation(F): FldNum → FldNum, MapOptimisedRepresentation(F): FldNum → FldNum, MapExample: Opt RepIntegralModel(F): FldNum → FldNum, MapExample: Integral Model
- Maximal Orders
- Creation of Elements
F ! a: FldNum, RngElt → FldNumEltelt< F | a >: FldNum, RngElt → FldNumEltF ! [a₀, a₁, ..., aₘ₋₁]: FldNum, [RngElt] → FldNumEltelt< F | [ a₀, a₁, ..., aₘ₋₁ ] >: FldNum, SeqEnum[RngElt] → FldNumEltelt< F | a₀, a₁, ..., aₘ₋₁>: FldNum, RngElt, ..., RngElt → FldNumEltRandom(F, m): FldNum, RngIntElt → FldNumEltExample: ElementsOne(K): FldNum → FldNumEltIdentity(K): FldNum → FldNumEltZero(K): FldNum → FldNumEltRepresentative(K): FldNum → FldNumElt
- Creation of Homomorphisms
hom< F -> R | r >: FldNum, Rng, RngElt → Maphom< F -> R | h, r >: FldNum, Rng, Map, RngElt → Maphom< F -> R | r >: FldNum, Rng, [RngElt] → Maphom< F -> R | h, r >: FldNum, Rng, Map, [RngElt] → MapExample: Homomorphisms
- Structure Operations
- General Functions
- Related Structures
GroundField(F): FldNum → FldBaseField(F): FldNum → FldCoefficientField(F): FldNum → FldCoefficientRing(F): FldNum → FldAbsoluteField(F): FldNum → FldNumSimpleExtension(F): FldNum → FldNumRelativeField(F, L): FldNum, FldNum → FldNumRelativeField(Q, L): FldRat, FldNum → FldNumComponents(F): FldNum → [FldNum]Example: CompositumPrimeRing(F): FldNum → RngRatPrimeField(F): FldNum → RngRatCentre(F): FldNum → FldNumEmbed(F, L, a): FldNum, FldNum, FldNumEltEmbed(F, L, a): FldNum, FldNum, [FldNumElt]EmbeddingMap(F, L): FldNum, FldNum → MapExample: emMinkowskiSpace(F): FldNum → Lat, MapCompletion(K, P): FldNum, RngOrdIdl → FldLoc, Mapcomp<K|P>: FldNum, RngOrdIdl → FldLoc, MapCompletion(K, P): FldNum, PlcNumElt → FldLoc, Map
- Representing Fields as Vector Spaces
Algebra(K, J): FldNum, Fld → AlgAss, MapAlgebra(K, J, S): FldNum, Fld, [FldNumElt] → AlgAss, MapVectorSpace(K, J): FldNum, Fld → ModTupFld, MapKSpace(K, J): FldNum, Fld → ModTupFld, MapVectorSpace(K, J, S): FldNum, Fld, [FldNumElt] → ModTupFld, MapKSpace(K, J, S): FldNum, Fld, [FldNumElt] → ModTupFld, MapExample: Vector Space Eg
- Invariants
- Basis Representation
- Ring Predicates
- Field Predicates
- Fields with a Labelled Embedding
- Creation Functions
EmbeddedNumberField(L, c): FldNum, FldComElt → BoolElt, FldNumEmbEmbeddedNumberField(L, c): FldNum, FldReElt → BoolElt, FldNumEmbEmbeddedNumberField(f, r): RngUPolElt, FldReElt → BoolElt, FldNumEmbEmbeddedNumberField(f, r): RngUPolElt, FldComElt → BoolElt, FldNumEmbEmbeddedNumberField(f, i): RngUPolElt, RngIntElt → FldNumEmbEmbeddedNumberField(L, i): FldNum, RngIntElt → FldNumEmbEmbeddedSplittingField(f): RngUPolElt → FldNumEmb, SeqEnumSubfield(L, K): FldNumEmb, FldNum → FldNumEmb
- Composition and Intersection
- Embedding and Reconstruction
- Element Operations
- Parent and Category
- Arithmetic
+ a: FldNumElt → FldNumElt- a: FldNumElt → FldNumElta + b: FldNumElt, FldNumElt → FldNumElta - b: FldNumElt, FldNumElt → FldNumElta * b: FldNumElt, FldNumElt → FldNumElta / b: FldNumElt, FldNumElt → FldNumElta ^ k: FldNumElt, RngIntElt → FldNumEltSqrt(a): FldNumElt → FldNumEltSquareRoot(a): FldNumElt → FldNumEltRoot(a, n): FldNumElt, RngIntElt → FldNumEltIsPower(a, k): FldNumElt, RngIntElt → BoolElt, FldNumEltIsSquare(a): FldNumElt → BoolElt, FldNumEltDenominator(a): FldNumElt → RngIntEltNumerator(a): FldNumElt → RngIntEltQround(E, M): FldNumElt, RngIntElt → FldNumElt
- Equality and Membership
- Predicates on Elements
- Field Generators
- Real and Complex Embeddings
- Heights
- Norm, Trace, and Minimal Polynomial
Norm(a): FldNumElt → FldNumEltNorm(a, R): FldNumElt, Rng → RngEltAbsoluteNorm(a): FldNumElt → FldRatEltNormAbs(a): FldNumElt → FldRatEltTrace(a): FldNumElt → FldNumEltTrace(a): FldNumElt → FldRatEltTrace(a, R): FldNumElt, Rng → RngEltAbsoluteTrace(a): FldNumElt → FldRatEltTraceAbs(a): FldNumElt → FldRatEltCharacteristicPolynomial(a): FldNumElt → RngUPolEltCharacteristicPolynomial(a, R): FldNumElt, Rng → RngUPolEltAbsoluteCharacteristicPolynomial(a): FldNumElt → RngUPolEltMinimalPolynomial(a): FldNumElt → RngUPolEltMinimalPolynomial(a, R): FldNumElt, Rng → RngUPolEltAbsoluteMinimalPolynomial(a): FldNumElt → RngUPolEltRepresentationMatrix(a): FldNumElt → NumMatEltRepresentationMatrix(a, R): FldNumElt, Rng → NumMatEltAbsoluteRepresentationMatrix(a): FldNumElt → NumMatEltExample: Norms Etc
- Other Functions
ElementToSequence(a): FldNumElt → [ FldNumElt ]Eltseq(a): FldNumElt → [ FldNumElt ]Eltseq(E, k): FldNumElt, FldNum → [RngElt]Eltseq(E, k): FldNumElt, Rng → [RngElt]Flat(e): FldNumElt → [ FldRatElt]a[i]: FldNumElt, RngIntElt → FldRatElta[i]: FldNumElt, RngIntElt → FldNumEltProductRepresentation(a): FldNumElt → [ FldNumElt ], [ RngIntElt ]ProductRepresentation(P, E): [ FldNumElt ], [ RngIntElt ] → FldNumEltPowerProduct(P, E): [FldNumElt], [RngIntElt] → FldNumElt
- Class Group and Unit Group
- Galois Theory
- Solving Norm Equations
- Places and Divisors
- Creation of Structures
- Operations on Structures
- Creation of Elements
Place(I): RngOrdIdl → PlcNumEltDecomposition(K, p): FldNum, RngIntElt → SeqEnumDecomposition(K, I): FldNum, Infty → SeqEnumDecomposition(K, p): FldNum, PlcNumElt → SeqEnumDecomposition(m, p): Map[FldRat, FldNum], RngIntElt → SeqEnum[<PlcNumElt, RngIntElt>]Decomposition(m, p): Map[FldNum, FldNum], PlcNumElt → SeqEnum[<PlcNumElt, RngIntElt>]InfinitePlaces(K): FldNum → SeqEnumInfinitePlaces(K): FldRat → SeqEnumInfinitePlaces(K): RngInt → SeqEnumDivisor(pl): PlcNumElt → DivNumEltDivisor(I): RngOrdFracIdl → DivNumEltDivisor(x): FldNumElt → DivNumEltRealPlaces(K): FldRat → [PlcNumElt]RealPlaces(K): FldNum → [PlcNumElt]
- 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 → RngOrdIdlEvaluate(x, p): FldNumElt, PlcNumElt → RngEltEvaluate(x, p): FldRatElt, PlcNumElt → RngEltEvaluate(x, p): RngIntElt, Infty → RngEltRealEmbeddings(a): FldNumElt → []RealSigns(a): FldNumElt → []IsReal(p): PlcNumElt → BoolEltIsComplex(p): PlcNumElt → BoolEltIsFinite(p): PlcNumElt → BoolEltIsInfinite(p): PlcNumElt → BoolElt, RngIntEltExtends(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
- Number Field Database
