[Next][Prev] [Right] [Left] [Up] [Index] [Root]

CLASS FIELD THEORY

 
Acknowledgements
 
Introduction
      Overview
      Magma
 
Creation
      Ray Class Groups
      Selmer groups
      Maps
      Abelian Extensions
      Binary Operations
 
Galois Module Structure
      Predicates
      Constructions
 
Conversion to Number Fields
 
Invariants
 
Automorphisms
 
Norm Equations
 
Attributes
      Orders
      Abelian Extensions
 
Group Theoretic Functions
      Generic Groups
 
Bibliography







DETAILS

 
Introduction

      Overview

      Magma
            Example FldAb_hilbert (H39E1)

 
Creation

      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

      Selmer groups
            pSelmerGroup(p, S) : RngIntElt, { RngOrdIdl } -> GrpAb, Map
            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

 
Galois Module Structure

      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)

 
Attributes

      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)

 
Group Theoretic Functions

      Generic Groups
            GenericGroup(X) : [] -> GrpFp, Map
            AddGenerator(G, x) : GrpFP, . -> BoolElt, GrpFP, Map
            FindGenerators(G) : GrpFP -> []

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Wed Apr 24 15:09:57 EST 2013