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.
Acknowledgements
Ray Class Groups
Creation of Class Fields
Properties of Class Fields
The Ring of Witt Vectors of Finite Length
The Ring of Twisted Polynomials
Creation of Twisted Polynomial Rings
Operations with the Ring of Twisted Polynomials
Creation of Twisted Polynomials
Operations with Twisted Polynomials
Analytic Theory
Related Functions
Enumeration of Places
Bibliography
Ray Class Groups
RayResidueRing(D) : DivFunElt -> GrpAb, Map
RayClassGroup(D) : DivFunElt -> GrpAb, Map
RayClassGroupDiscLog(y, D) : DivFunElt, DivFunElt -> GrpAbElt
Example FldFunAb_classfield-structures (H43E1)
Creation of Class Fields
AbelianExtension(D, U) : DivFunElt, GrpAb -> FldFunAb
MaximalAbelianSubfield(K) : FldFunG -> FldFunAb
HilbertClassField(K, p) : FldFun, PlcFunElt -> FldFunAb
FunctionField(A) : FldFunAb -> FldFun
MaximalOrderFinite(A) : FldFunAb -> RngFunOrd
Example FldFunAb_classfields (H43E2)
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]
Unity(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
The Ring of Twisted Polynomials
Creation of Twisted Polynomial Rings
TwistedPolynomials(R) : Rng -> RngUPolTwst
Operations with the Ring of Twisted Polynomials
Unity(R) : RngUPolTwst -> RngUPolTwstElt
Zero(R) : RngUPolTwst -> RngUPolTwstElt
R eq S : RngUPolTwst, RngUPolTwst -> BoolElt
BaseRing(R) : RngUPolTwst -> Rng
R . i : RngUPolTwst, RngIntElt -> RngUPolTwstElt
Creation of Twisted Polynomials
AdditivePolynomialFromRoots(x, P) : RngElt, PlcFunElt -> RngUPolTwstElt
Random(F, n) : RngUPolTwst, RngIntElt -> RngUPolTwstElt
Example FldFunAb_additive-polynomial (H43E3)
Operations with Twisted Polynomials
LeadingCoefficient(F) : RngUPolTwstElt -> RngElt
ConstantCoefficient(F) : RngUPolTwstElt -> RngElt
Degree(F) : RngUPolTwstElt -> RngIntElt
Quotrem(F, G) : RngUPolTwstElt, RngUPolTwstElt -> RngUPolTwstElt, RngUPolTwstElt
GCD(F, G) : RngUPolTwstElt, RngUPolTwstElt -> RngUPolTwstElt
BaseRing(F) : RngUPolTwstElt -> Rng
Polynomial(G) : RngUPolTwstElt -> RngUPolElt
SpecialEvaluate(F, x) : RngUPolTwstElt, RngElt -> RngElt
SpecialEvaluate(F, x) : RngUPolElt, Any -> RngElt
Eltseq(F) : RngUPolTwstElt -> [RngElt]
Analytic Theory
CarlitzModule(R, x) : RngUPolTwst, RngUPolElt -> RngUPolTwstElt
Example FldFunAb_carlitz-module (H43E4)
AnalyticDrinfeldModule(F, p) : FldFun, PlcFunElt -> RngUPolTwstElt
Extend(D, x, p) : RngUPolTwstElt, RngElt, PlcFunElt -> RngUPolTwstElt
Example FldFunAb_drinfeld (H43E5)
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
Example FldFunAb_strong-approximation (H43E6)
NonSpecialDivisor(m): DivFunElt -> DivFunElt, RngIntElt
NormGroup(F) : FldFun -> DivFunElt, GrpAb
Sign(a, p) : FldFunElt, PlcFunElt -> RngElt
ChangeModel(F, p) : FldFun, PlcFunElt -> FldFun
Enumeration of Places
PlaceEnumInit(K) : FldFun -> PlcEnum
PlaceEnumInit(P) : PlcFunElt -> PlcEnum
PlaceEnumInit(K, Pos) : FldFun, [RngIntElt]) -> PlcEnum
PlaceEnumCopy(R) : PlcEnum -> PlcEnum
PlaceEnumPosition(R) : PlcEnum -> [RngIntElt]
PlaceEnumNext(R) : PlcEnum -> PlcFunElt
PlaceEnumCurrent(R) : PlcEnum -> PlcFunElt
Bibliography
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