CLASS FIELD THEORY

DETAILS

 
Introduction

      Overview

      Magma
            Example FldAb_hilbert (H56E1)

 
Creation

      Ray Class Groups
            RayClassGroup(I) : RngOrdIdl -> GrpAb, Map
            Example FldAb_ideal-ray (H56E2)
            RayResidueRing(I) : RngOrdIdl -> GrpAb, Map
            pSelmerGroup(p, S) : prime p, { RngOrdIdl } -> G, m
            Example FldAb_Selmer-group (H56E3)

      Maps
            InducedMap(m1, m2, h, c) : Map, Map, Map, RngIntElt -> Map
            InducedAutomorphism(r, h, c) : Map, Map, RngIntElt -> Map
            Example FldAb_inducedMap (H56E4)

      Abelian Extensions
            RayClassField(m) : Map -> FldAb
            AbelianExtension(I) : RngOrdIdl -> FldAb
            AbelianpExtension(m, p) : Map, RngIntElt -> FldAb
            Example FldAb_class-field (H56E5)
            AbelianExtension(I, P) : RngOrdIdl, [RngIntElt] -> FldAb
            HilbertClassField(K) : FldAlg -> FldAb
            MaximalAbelianSubfield(M) : RngOrd -> FldAb
            AbelianExtension(K) : FldOrd -> FldAb
            Example FldAb_hilbert-class-field (H56E6)

 
Galois Module Structure

      Predicates
            IsAbelian(A) : FldAb -> BoolElt
            IsNormal(A) : FldAb -> BoolElt
            IsCentral(A) : FldAb -> BoolElt

      Constructions
            GenusField(A): FldAb -> FldAb

 
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]
      Conductor(A) : FldAb -> RngOrdIdl, [RngIntElt]
      Degree(A) : FldAb -> RngIntElt
      AbsoluteDegree(A) : FldAb -> RngIntElt
      CoeffientField(A) : FldAb -> Field
      BaseRing(A) : FldAb -> Ring
      NormGroup(A) : FldAb -> Map, RngOrdIdl, [RngIntElt]
      DecompositionField(p, A) : RngOrdIdl, FldAb -> FldAb
      DecompositionField(p, A) : PlcNumElt, FldAb -> FldAb
      DecompositionGroup(p, A) : RngOrdIdl, 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
      FrobeniusAutomosphism(A, p) : FldAb, RngOrdIdl -> Map
      AutomorphismGroup(A) : FldAb -> GrpFP, [Map], Map

 
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 (H56E7)

 
Attributes

      Orders
            o`CyclotomicExtensions : RngOrd -> [Rec]
            Example FldAb_cyclotomic-extension (H56E8)

      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 (H56E9)

 
Group Theoretic Functions

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

 
Bibliography