Class Field Theory
- Introduction
- Creation
- Ray Class Groups
RayClassGroup(I): RngOrdIdl → GrpAb, MapRayClassGroup(I, T): RngOrdIdl, SeqEnum[RngIntElt] → GrpAb, MapRayClassGroup(D): DivNumElt → GrpAb, MapRayClassGroup(P): PlcNumElt → GrpAb, MapExample: Ideal RayRayResidueRing(I): RngOrdIdl → GrpAb, MapRayResidueRing(I, T): RngOrdIdl, SeqEnum[RngIntElt] → GrpAb, MapRayResidueRing(D): DivNumElt → GrpAb, MapRayResidueRing(P): PlcNumElt → GrpAb, Map
- Selmer Groups
- Maps
InducedMap(m1, m2, h, c): Map, Map, Map, RngIntElt → MapInducedAutomorphism(r, h, c): Map, Map, RngIntElt → MapExample: induced Map
- Abelian Extensions
RayClassField(m): Map → FldAbAbelianExtension(m): Map → FldAbRayClassField(m, I, T): Map, RngOrdIdl, [RngIntElt] → FldAbAbelianExtension(m, I, T): Map, RngOrdIdl, [RngIntElt] → FldAbRayClassField(m, I): Map, RngOrdIdl → FldAbAbelianExtension(m, I): Map, RngOrdIdl → FldAbAbelianExtension(I): RngOrdIdl → FldAbRayClassField(D): DivNumElt → FldAbRayClassField(P): PlcNumElt → FldAbAbelianpExtension(m, p): Map, RngIntElt → FldAbExample: Class FieldAbelianExtension(I, P): RngOrdIdl, [RngIntElt] → FldAbHilbertClassField(K): FldAlg → FldAbMaximalAbelianSubfield(M): RngOrd → FldAbMaximalAbelianSubfield(F): FldOrd → FldAbMaximalAbelianSubfield(K): FldNum → FldAbAbelianExtension(K): FldAlg → FldAbAbelianExtension(M): RngOrd → FldAbExample: Hilbert Class Field
- Binary Operations
- Galois Module Structure
- Conversion to Number Fields
- Invariants
Discriminant(A): FldAb → RngOrdIdl, [RngIntElt]AbsoluteDiscriminant(A): FldAb → RngIntEltConductor(A): FldAb → RngOrdIdl, [RngIntElt]Degree(A): FldAb → RngIntEltAbsoluteDegree(A): FldAb → RngIntEltCoefficientRing(A): FldAb → FldCoefficientField(A): FldAb → FldBaseField(A): FldAb → FldBaseRing(A): FldAb → RngCoefficientRing(A): FldAb → RngNormGroup(A): FldAb → Map, RngOrdIdl, [RngIntElt]DecompositionField(p, A): RngOrdIdl, FldAb → FldAbDecompositionField(p, A): PlcNumElt, FldAb → FldAbDecompositionGroup(p, A): RngIntElt, FldAb → GrpAbDecompositionGroup(p, A): RngOrdIdl, FldAb → GrpAbDecompositionGroup(p, A): PlcNumElt, FldAb → GrpAbDecompositionType(A, p): FldAb, RngOrdIdl → [Tpl]DecompositionType(A, p): FldAb, PlcNumElt → [Tpl]DecompositionType(A, p): FldAb, RngIntElt → [Tpl]DecompositionTypeFrequency(A, l): FldAb, [ ] → MsetDecompositionTypeFrequency(A, a, b): FldAb, RngIntElt, RngIntElt → Mset
- Automorphisms
ArtinMap(A): FldAb → MapFrobeniusAutomorphism(A, p): FldAb, RngOrdIdl → MapAutomorphismGroup(A): FldAb → GrpFP, [Map], MapProbableAutomorphismGroup(A): FldAb → GrpFP, SeqEnumImproveAutomorphismGroup(F, E): FldAb, SeqEnum → GrpFP, SeqEnumExample: Probable Automorphism GroupAbsoluteGaloisGroup(A): FldAb → GrpPerm, SeqEnum, GaloisDataTwoCocycle(A): FldAb → UserProgram
- Norm Equations
IsLocalNorm(A, x, p): FldAb, RngOrdElt, RngOrdIdl → BoolEltIsLocalNorm(A, x, p): FldAb, RngIntElt, RngInt → BoolEltIsLocalNorm(A, x, i): FldAb, RngOrdElt, RngIntElt → BoolEltIsLocalNorm(A, x, p): FldAb, RngOrdElt, PlcNumElt → BoolEltIsLocalNorm(A, x): FldAb, RngOrdElt → BoolEltKnot(A): FldAb → GrpAbNormEquation(A, x): FldAb, RngOrdElt → BoolElt, [RngOrdElt]IsNorm(A, x): FldAb, RngOrdElt → BoolEltExample: Norm Equation
- Attributes
- Group Theoretic Functions
