Magma
Example FldAb_hilbert (H39E1)
Ray Class Groups
RayClassGroup(I) : RngOrdIdl -> GrpAb, Map
RayClassGroup(D) : DivNumElt -> GrpAb, Map
Example FldAb_ideal-ray (H39E2)
RayResidueRing(I) : RngOrdIdl -> GrpAb, Map
RayResidueRing(D) : DivNumElt -> GrpAb, Map
pSelmerGroup(p, S) : RngIntElt, { RngOrdIdl } -> G, m
Example FldAb_Selmer-group (H39E3)
Maps
InducedMap(m1, m2, h, c) : Map, Map, Map, RngIntElt -> Map
InducedAutomorphism(r, h, c) : Map, Map, RngIntElt -> Map
Example FldAb_inducedMap (H39E4)
Abelian Extensions
RayClassField(m) : Map -> FldAb
AbelianExtension(I) : RngOrdIdl -> FldAb
RayClassField(D) : DivNumElt -> FldAb
AbelianpExtension(m, p) : Map, RngIntElt -> FldAb
Example FldAb_class-field (H39E5)
AbelianExtension(I, P) : RngOrdIdl, [RngIntElt] -> FldAb
HilbertClassField(K) : FldAlg -> FldAb
MaximalAbelianSubfield(M) : RngOrd -> FldAb
AbelianExtension(K) : FldAlg -> FldAb
Example FldAb_hilbert-class-field (H39E6)
Binary Operations
A eq B : FldAb, FldAb -> BoolElt
A subset B : FldAb, FldAb -> BoolElt
A * B : FldAb, FldAb -> FldAb
A meet B : FldAb, FldAb -> FldAb
Predicates
IsAbelian(A) : FldAb -> BoolElt
IsNormal(A) : FldAb -> BoolElt
IsCentral(A) : FldAb -> BoolElt
Constructions
GenusField(A): FldAb -> FldAb
H2_G_A(A) : FldAb -> ModTupRng
NormalSubfields(A) : FldAb -> []
AbelianSubfield(A, U) : FldAb, GrpAb -> FldAb
CohomologyModule(A) : FldAb -> ModGrp, Map, Map, Map
Conversion to Number Fields
EquationOrder(A) : FldAb -> RngOrd
NumberField(A) : FldAb -> FldNum
MaximalOrder(A) : FldAb -> RngOrd
Components(A) : FldAb -> [RngOrd]
Generators(A) : FldAb -> [ ], [ ], [ ]
Invariants
Discriminant(A) : FldAb -> RngOrdIdl, [RngIntElt]
AbsoluteDiscriminant(A) : FldAb -> RngIntElt
Conductor(A) : FldAb -> RngOrdIdl, [RngIntElt]
Degree(A) : FldAb -> RngIntElt
AbsoluteDegree(A) : FldAb -> RngIntElt
CoefficientRing(A) : FldAb -> Fld
BaseRing(A) : FldAb -> Rng
NormGroup(A) : FldAb -> Map, RngOrdIdl, [RngIntElt]
DecompositionField(p, A) : RngOrdIdl, FldAb -> FldAb
DecompositionField(p, A) : PlcNumElt, FldAb -> FldAb
DecompositionGroup(p, A) : RngIntElt, FldAb -> GrpAb
DecompositionGroup(p, A) : PlcNumElt, FldAb -> GrpAb
DecompositionType(A, p) : FldAb, RngOrdIdl -> [Tpl]
DecompositionType(A, p) : FldAb, PlcNumElt -> [Tpl]
DecompositionType(A, p) : FldAb, RngIntElt -> [Tpl]
DecompositionTypeFrequency(A, l) : FldAb, [ ] -> Mset
DecompositionTypeFrequency(A, a, b) : FldAb, RngIntElt, RngIntElt -> Mset
Automorphisms
ArtinMap(A) : FldAb -> Map
FrobeniusAutomorphism(A, p) : FldAb, RngOrdIdl -> Map
AutomorphismGroup(A) : FldAb -> GrpFP, [Map], Map
ProbableAutomorphismGroup(A) : FldAb -> GrpFP, SeqEnum
ImproveAutomorphismGroup(F, E) : FldAb, SeqEnum -> GrpFP, SeqEnum
Example FldAb_ProbableAutomorphismGroup (H39E7)
AbsoluteGaloisGroup(A) : FldAb -> GrpPerm, SeqEnum, GaloisData
TwoCocycle(A) : FldAb -> UserProgram
Norm Equations
IsLocalNorm(A, x, p) : FldAb, RngOrdElt, RngOrdIdl -> BoolElt
IsLocalNorm(A, x, i) : FldAb, RngOrdElt, RngIntElt -> BoolElt
IsLocalNorm(A, x, p) : FldAb, RngOrdElt, PlcNumElt -> BoolElt
IsLocalNorm(A, x) : FldAb, RngOrdElt -> BoolElt
Knot(A) : FldAb -> GrpAb
NormEquation(A, x) : FldAb, RngOrdElt -> BoolElt, [RngOrdElt]
IsNorm(A, x) : FldAb, RngOrdElt -> BoolElt
Example FldAb_norm-equation (H39E8)
Orders
o`CyclotomicExtensions : RngOrd -> [Rec]
Example FldAb_cyclotomic-extension (H39E9)
Abelian Extensions
A`Components : FldAb -> [Rec]
A`DefiningGroup : FldAb -> Rec
A`IsAbelian : FldAb -> Bool
A`IsNormal : FldAb -> Bool
A`IsCentral : FldAb -> Bool
Example FldAb_abelian-extension-attributes (H39E10)
Generic Groups
GenericGroup(X) : [] -> GrpFp, Map
AddGenerator(G, x) : GrpFP, . -> BoolElt, GrpFP, Map
FindGenerators(G) : GrpFP -> []
Bibliography
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Mon Dec 17 14:40:36 EST 2012