Global function fields admit a class field theory in
the same way
as number fields do (Chapter CLASS FIELD THEORY). From a computational point of
view the main difference is the use of divisors rather than
ideals and the availability in general of analytical methods; see
Section Analytic Theory.
Class field theory deals with the abelian extensions of a given
field. In the number field case, all abelian extensions can be parameterized
using more general class groups, in the case of global function fields,
the same will be achieved using the divisor class group and extensions of
it.
- Ray Class Groups
- Creation of Class Fields
- Properties of Class Fields
- Conductor(m) : DivFunElt -> DivFunElt
- Conductor(m, U) : DivFunElt, GrpAb -> DivFunElt
- Conductor(A) : FldFunAb -> DivFunElt
- DiscriminantDivisor(m, U) : DivFunElt, GrpAb -> DivFunElt
- DiscriminantDivisor(A) : FldFunAb -> DivFunElt
- DegreeOfExactConstantField(m) : DivFunElt -> RngIntElt
- DegreeOfExactConstantField(m, U) : DivFunElt, GrpAb -> RngIntElt
- DegreeOfExactConstantField(A) : FldFunAb -> RngIntElt
- Genus(m, U) : DivFunElt, GrpAb -> RngIntElt
- Genus(A) : FldFunAb -> RngIntElt
- DecompositionType(m, U, p) : DivFunElt, GrpAb, PlcFunElt -> [<f,e>]
- DecompositionType(A, p) : FldFunAb, PlcFunElt -> [<f,e>]
- NumberOfPlacesOfDegreeOne(m, U) : DivFunElt, GrpAb -> RngIntElt
- NumberOfPlacesOfDegreeOne(A) : FldFunAb -> RngIntElt
- Degree(A) : FldFunAb -> RngIntElt
- BaseField(A) : FldFunAb -> FldFunG
- A eq B : FldFunAb, FldFunAb -> BoolElt
- A subset B : FldFunAb, FldFunAb -> BoolElt
- A meet B : FldFunAb, FldFunAb -> FldFunAb
- A * B : FldFunAb, FldFunAb -> FldFunAb
- The Ring of Witt Vectors of Finite Length
- WittRing(F, n) : Fld, RngIntElt -> RngWitt
- W ! a : RngWitt, . -> RngWittElt
- BaseRing(W) : RngWitt -> Fld
- Length(W) : RngWitt -> RngIntElt
- Eltseq(a) : RngWittElt -> [FldElt]
- One(W) : RngWitt -> RngWittElt
- W . 1 : RngWitt, RngIntElt -> RngWittElt
- FrobeniusMap(W) : RngWitt -> Map
- FrobeniusImage(e) : RngWittElt -> RngWittElt
- VerschiebungMap(W) : RngWitt -> Map
- VerschiebungImage(e) : RngWittElt -> RngWittElt
- Random(W) : RngWitt -> RngWittElt
- Random(W, n) : RngWitt, RngIntElt -> RngWittElt
- TeichmuellerSystem(R) : Any -> [RngLocElt]
- LocalRing(W) : RngWitt -> RngLoc, Map
- ArtinSchreierMap(W) : RngWitt -> Map
- ArtinSchreierImage(e) : RngWittElt -> RngWittElt
- FunctionField(e) : RngWittElt -> FldFun, Map
- MaximalOrderFinite(u) : RngWittElt -> RngFunOrd
- SMaximalOrder(u, S) : RngWittElt, [PlcFunElt] -> RngFunOrd
- MaximalOrders(u) : RngWittElt -> RngFunOrd, RngFunOrd
- The Ring of Twisted Polynomials
- Analytic Theory
- CarlitzModule(R, x) : RngUPolTwst, RngUPolElt -> RngUPolTwstElt
- Example FldFunAb_carlitz-module (H46E5)
- AnalyticDrinfeldModule(F, p) : FldFun, PlcFunElt -> RngUPolTwstElt
- Extend(D, x, p) : RngUPolTwstElt, RngElt, PlcFunElt -> RngUPolTwstElt
- Example FldFunAb_drinfeld (H46E6)
- Exp(x, p) : RngElt, PlcFunElt -> RngUPolTwstElt
- AnalyticModule(x, p) : RngElt, PlcFunElt -> RngElt
- CanNormalize(F) : RngUPolTwstElt -> BoolElt, RngUPolTwstElt, RngElt
- CanSignNormalize(F) : RngUPolTwstElt -> BoolElt, RngUPolTwstElt, RngElt
- AlgebraicToAnalytic(F, p) : RngUPolTwstElt, PlcFunElt -> RngUPolTwstElt
- Related Functions
- StrongApproximation(m, S): DivFunElt, [<PlcFunElt, FldFunElt>] -> FldFunElt
- StrongApproximation(S, Z, V): [PlcFunElt], [FldFunGElt], [RngIntElt] -> FldFunElt
- ChineseRemainderTheorem(S, Z, V): [PlcFunElt], [FldFunGElt], [RngIntElt] -> FldFunElt
- Example FldFunAb_strong-approximation (H46E7)
- NonSpecialDivisor(m): DivFunElt -> DivFunElt, RngIntElt
- NormGroup(F) : FldFun -> DivFunElt, GrpAb
- Sign(a, p) : FldFunElt, PlcFunElt -> RngElt
- ChangeModel(F, p) : FldFun, PlcFunElt -> FldFun
- ArtinSchreierReduction(u, P): FldFunGElt, PlcFunElt -> RngIntElt, FldFunElt
- Enumeration of Places
- Bibliography
V2.28, 13 July 2023