Creation of Algebraic Function Fields and their Orders
Creation of Algebraic Function Fields
ext< K | f > : FldFunRat, RngUPolElt -> FldFun
FunctionField(f : parameters) : RngMPolElt -> FldFun
FunctionField(S) : [RngUPolElt] -> FldFun
HermitianFunctionField(p, d) : RngIntElt, RngIntElt -> FldFun
AssignNames(~F, s) : FldFun, [ MonStgElt ] ->
FunctionField(R) : Rng -> FldFunG
Example FldFunG_Creation (H55E1)
Example FldFunG_creation-rel (H55E2)
Example FldFunG_creation-non-simple (H55E3)
Example FldFunG_creation_herm (H55E4)
Creation of Orders of Algebraic Function Fields
EquationOrderFinite(F) : FldFun -> RngFunOrd
MaximalOrderFinite(F) : FldFun -> RngFunOrd
EquationOrderInfinite(F) : FldFun -> RngFunOrd
MaximalOrderInfinite(F) : FldFun -> RngFunOrd
IntegralClosure(R, F) : Rng, FldFun -> RngFunOrd
EquationOrder(O) : RngFunOrd -> RngFunOrd
MaximalOrder(O) : RngFunOrd -> RngFunOrd
SetOrderMaximal(O, b) : RngFunOrd, BoolElt ->
ext<O | f> : RngFunOrd, RngUPolElt -> RngFunOrd
Example FldFunG_orders (H55E5)
Example FldFunG_int_cl (H55E6)
Order(O, T, d) : RngFunOrd, AlgMatElt, RngElt -> RngFunOrd
Order(O, M) : RngFunOrd, ModDed -> RngFunOrd
Order(O, S) : RngFunOrd, [FldFunElt] -> RngFunOrd
Simplify(O) : RngFunOrd -> RngFunOrd
O1 + O2 : RngFunOrd, RngFunOrd -> RngFunOrd
Example FldFunG_order-create-more (H55E7)
Orders and Ideals
MultiplicatorRing(I) : RngFunOrdIdl -> RngFunOrd
pMaximalOrder(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrd
pRadical(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl
Other Related Structures
PrimeRing(F) : FldFun -> Rng
ConstantField(F) : FldFun -> Rng
ExactConstantField(F) : FldFunG -> Rng, Map
BaseRing(F) : FldFun -> Rng
ISABaseField(F,G) : Fld, Fld -> BoolElt
BaseRing(O) : RngFunOrd -> Rng
BaseRing(FF) : FldFunOrd -> Rng
SubOrder(O) : RngFunOrd -> RngFunOrd
FunctionField(O) : RngFunOrd -> FldFun
FieldOfFractions(O) : RngFunOrd -> FldFunOrd
Order(FF) : FldFunOrd -> RngFunOrd
RationalExtensionRepresentation(F) : FldFunG -> FldFun
AbsoluteOrder(O) : RngFunOrd -> RngFunOrd
UnderlyingRing(F) : FldFunG -> FldFunG
Places(F) : FldFun -> PlcFun
DivisorGroup(F) : FldFun -> DivFun
DifferentialSpace(F) : FldFun -> DiffFun
Example FldFunG_related-structures (H55E8)
Example FldFunG_related-structures-rat-ext (H55E9)
WeilRestriction(E, n) : FldFun, RngIntElt -> FldFun, UserProgram
ConstantFieldExtension(F, E) : FldFun, Rng -> FldFun, Map
Example FldFunG_cfe (H55E10)
Reduce(O) : RngFunOrd -> RngFunOrd
General Structure Invariants
Characteristic(F) : FldFun -> RngIntElt
IsPerfect(F) : Fld -> BoolElt
Degree(F) : FldFun -> RngIntElt
AbsoluteDegree(F) : FldFun -> RngIntElt
DefiningPolynomial(F) : FldFun -> RngUPolElt
DefiningPolynomials(F) : FldFun -> [RngUPolElt]
Basis(F) : FldFun -> SeqEnum[FldFunElt]
TransformationMatrix(O1, O2) : RngFunOrd, RngFunOrd -> AlgMatElt, RngElt
BasisMatrix(O) : RngFunOrd -> AlgMatElt
PrimitiveElement(O) : RngFunOrd -> RngFunOrdElt
Discriminant(O) : RngFunOrd -> .
AbsoluteDiscriminant(O) : RngFunOrd -> .
DimensionOfExactConstantField(F) : FldFun -> RngIntElt
Genus(F) : FldFun -> RngIntElt
Example FldFunG_invar (H55E11)
Example FldFunG_invar-non-simple (H55E12)
GapNumbers(F) : FldFunG -> SeqEnum[RngIntElt]
GapNumbers(F, P) : FldFunG, PlcFunElt -> SeqEnum[RngIntElt]
SeparatingElement(F) : FldFunG -> FldFunGElt
RamificationDivisor(F) : FldFunG -> DivFunElt
WeierstrassPlaces(F) : FldFunG -> [PlcFunElt]
WronskianOrders(F) : FldFunG -> [RngIntElt]
Different(O) : RngFunOrd -> RngFunOrdIdl
Index(O, S) : RngFunOrd, RngFunOrd -> Any
Galois Groups
GaloisGroup(f) : RngUPolElt -> GrpPerm, [ FldPrElt ], Any
Example FldFunG_GaloisGroups (H55E13)
Example FldFunG_GaloisGroups2 (H55E14)
Subfields
Subfields(F) : FldFun -> SeqEnum[FldFun]
Example FldFunG_Subfields (H55E15)
Automorphisms over the Base Field
Automorphisms(K, k) : FldFun, FldFun -> [Map]
AutomorphismGroup(K, k) : FldFun, FldFun -> GrpFP, Map
Example FldFunG_Automorphisms (H55E16)
IsSubfield(K, L) : FldFun, FldFun -> BoolElt, Map
IsIsomorphicOverQt(K, L) : FldFun, FldFun -> BoolElt, Map
Example FldFunG_IsSubfield (H55E17)
General Automorphisms
Isomorphisms(K, E) : FldFunG, FldFunG -> [Map]
IsIsomorphic(K, E) : FldFunG, FldFunG -> BoolElt, Map
Automorphisms(K) : FldFunG -> [Map]
Isomorphisms(K,E,p1,p2) : FldFunG, FldFunG, PlcFunElt, PlcFunElt -> [Map]
AutomorphismGroup(K) : FldFunG -> GrpFP, Map
AutomorphismGroup(K,f) : FldFunG, Map -> Grp, Map, SeqEnum
Field Morphisms
IsMorphism(f) : Map -> Bool
FieldMorphism(f) : Map -> Map
IdentityFieldMorphism(F) : Fld -> Map
IsIdentity(f) : Map -> BoolElt
Equality(f, g) : Map, Map -> Bool
HasInverse(f) : Map -> MonStgElt, Map
Composition(f, g) : Map, Map -> Map
Example FldFunG_Isomorphisms (H55E18)
Functions relative to the Exact Constant Field
NumberOfPlacesOfDegreeOverExactConstantField(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOneOverExactConstantField(F) : FldFun -> RngIntElt
NumberOfPlacesOfDegreeOneOverExactConstantField(F, m) : FldFun -> RngIntElt
SerreBound(F) : FldFun -> RngIntElt
IharaBound(F) : FldFun -> RngIntElt
NumberOfPlacesOfDegreeOneECFBound(F) : FldFun -> RngIntElt
LPolynomial(F) : FldFun -> RngUPolElt
LPolynomial(F, m) : FldFun, RngIntElt -> RngUPolElt
ZetaFunction(F) : FldFun -> FldFunRatUElt
ZetaFunction(F, m) : FldFun, RngIntElt -> FldFunRatUElt
Functions Relative to the Constant Field
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
HasPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
RandomPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
Example FldFunG_global-function-fields (H55E19)
Example FldFunG_global1 (H55E20)
Functions related to Class Group
UnitRank(O) : RngFunOrd -> RngIntElt
UnitGroup(O) : RngFunOrd -> GrpAb, Map
Regulator(O) : RngFunOrd -> RngIntElt
PrincipalIdealMap(O) : RngFunOrd -> Map
Example FldFunG_global-class-ex (H55E21)
ClassGroup(F : parameters) : FldFun -> GrpAb, Map, Map
ClassGroup(O) : RngFunOrd -> GrpAb, Map, Map
ClassGroupExactSequence(O) : RngFunOrd -> Map, Map, Map
ClassGroupAbelianInvariants(F : parameters) : FldFun -> SeqEnum
ClassGroupAbelianInvariants(O) : RngFunOrd -> SeqEnum
ClassNumber(F) : FldFun -> RngIntElt
ClassNumber(O) : RngFunOrd -> RngIntElt
Example FldFunG_class-group (H55E22)
GlobalUnitGroup(F) : FldFun -> GrpAb, Map
ClassGroupPRank(F) : FldFunG -> RngIntElt
HasseWittInvariant(F) : FldFunG -> RngIntElt
IndependentUnits(O) : RngFunOrd -> SeqEnum[RngFunOrdElt]
FundamentalUnits(O) : RngFunOrd -> SeqEnum[RngFunOrdElt]
Example FldFunG_orders (H55E23)
Structure Predicates
IsGlobal(F) : FldFun -> BoolElt
IsRationalFunctionField(F) : FldFunG -> BoolElt
IsFiniteOrder(O) : RngFunOrd -> BoolElt
IsEquationOrder(O) : RngFunOrd -> BoolElt
IsAbsoluteOrder(O) : RngFunOrd -> BoolElt
IsMaximal(O) : RngFunOrd -> BoolElt
IsTamelyRamified(O) : RngFunOrd -> BoolElt
IsTotallyRamified(O) : RngFunOrd -> BoolElt
IsUnramified(O) : RngFunOrd -> BoolElt
IsWildlyRamified(O) : RngFunOrd -> BoolElt
Homomorphisms
hom<F -> R | g> : FldFun, Rng, RngElt -> Map
hom< O -> R | g > : RngFunOrd, Rng, RngElt -> Map
IsRingHomomorphism(m) : Map -> BoolElt
Example FldFunG_hom (H55E24)
hom< O -> R | b_1, ..., b_n > : RngFunOrd, Rng, RngElt, ..., RngElt -> Map
Creation of Elements
F . 1 : FldFun -> FldFunElt
Name(F, i) : FldFun, RngIntElt -> FldFunElt
O . i : RngFunOrd, RngIntElt -> FldFunOrdElt
F ! a : FldFun, . -> FldFunElt
O ! a : RngFunOrd, . -> RngFunOrdElt
FF ! a : FldFunOrd, Any -> FldFunOrdElt
elt< F | a_0, a_1, ..., a_(n - 1)> : FldFun, RngElt , ..., RngElt -> FldFunElt
elt< O | a_1, a_2, ..., a_(n)> : RngFunOrd, RngElt , ..., RngElt -> RngFunOrdElt
Random(F, m) : FldFun, RngIntElt -> FldFunElt
Sequence Conversions
ElementToSequence(a) : FldFunElt -> SeqEnum[FldElt]
F ! [ a_0, a_1, ..., a_(n - 1) ] : FldFun, SeqEnum -> FldFunElt
O ! [ a_1, a_2, ..., a_(n) ] : RngFunOrd, SeqEnum -> RngFunOrdElt
Example FldFunG_Elements (H55E25)
Arithmetic Operators
Modexp(a, k, m) : RngFunOrdElt, RngIntElt, RngUPolElt -> RngFunOrdElt
a mod I : RngFunOrdElt, RngFunOrdIdl -> RngFunOrdElt
Modinv(a, m) : RngFunOrdElt, RngFunOrdIdl -> RngFunOrdElt
Predicates on Elements
IsDivisibleBy(a, b) : FldFunElt, FldFunElt -> BoolElt, FldFunElt
IsSeparating(a) : FldFunGElt -> BoolElt
IsConstant(a) : FldFunElt -> BoolElt, RngElt
IsGlobalUnit(a) : FldFunElt -> BoolElt
IsGlobalUnitWithPreimage(a) : FldFunElt -> BoolElt, GrpAbElt
IsUnitWithPreimage(a) : RngFunOrdElt -> BoolElt, GrpAbElt
Functions related to Norm and Trace
RepresentationMatrix(a) : FldFunGElt -> AlgMatElt
Trace(a, R) : FldFunElt, Rng -> RngElt
Norm(a, R) : FldFunElt, Rng -> RngElt
Example FldFunG_elements-norm-trace (H55E26)
Functions related to Orders and Integrality
IntegralSplit(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt, RngElt
Numerator(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt
Numerator(a) : FldFunOrdElt -> RngFunOrdElt
Denominator(a, O) : FldFunElt, RngFunOrd -> RngElt
Denominator(a) : FldFunOrdElt -> RngElt
Minimum(a, O) : FldFunElt, RngFunOrd -> RngElt, RngElt
Functions related to Places and Divisors
Evaluate(a, P) : FldFunElt, PlcFunElt -> RngElt
Lift(a, P) : RngElt, PlcFunElt -> FldFunElt
Valuation(a, P) : FldFunElt, PlcFunElt -> RngIntElt
Expand(a, P) : FldFunGElt, PlcFunElt -> RngSerElt, FldFunGElt
Divisor(a) : FldFunGElt -> DivFunElt
Zeros(a) : FldFunGElt -> [PlcFunElt]
Zeros(F, a) : FldFunG, FldFunGElt -> [PlcFunElt]
Poles(a) : FldFunElt -> SeqEnum[PlcFunElt]
Poles(F, a) : FldFun, FldFunGElt -> [PlcFunElt]
Degree(a) : FldFunElt -> RngIntElt
CommonZeros(L) : [FldFunGElt] -> [PlcFunElt]
CommonZeros(F, L) : FldFunG, SeqEnum[ FldFunGElt ] -> SeqEnum[ PlcFunElt ]
Example FldFunG_elements (H55E27)
Module(L, R) : SeqEnum[ FldFunGElt ], Rng -> Mod, Map, SeqEnum[ ModElt ]
Relations(L, R) : SeqEnum[ FldFunElt ], Rng -> ModTupRng
Roots(f, D) : RngUPolElt, DivFunElt -> SeqEnum[ FldFunElt ]
Example FldFunG_module (H55E28)
Other Operations on Elements
ProductRepresentation(a) : FldFunGElt -> [FldFunGElt], [RngIntElt]
ProductRepresentation(Q, S) : [FldFunGElt], [RngIntElt] -> FldFunGElt
RationalFunction(a) : FldFunGElt -> RngElt
Differentiation(x, a) : FldFunGElt, FldFunGElt -> FldFunGElt
Differentiation(x, n, a) : FldFunGElt, RngIntElt, FldFunGElt -> FldFunGElt
DifferentiationSequence(x, n, a) : FldFunGElt, RngIntElt, FldFunGElt -> SeqEnum
PrimePowerRepresentation(x, k, a) : FldFunGElt, RngIntElt, FldFunGElt -> SeqEnum
Different(a) : RngFunOrdElt -> RngFunOrdElt
Example FldFunG_elements-other_ops (H55E29)
Creation of Ideals
ideal< O | a_1, a_2, ... , a_m > : RngFunOrd, RngElt, ..., RngElt -> RngFunOrdIdl
ideal< O | T, d > : RngFunOrd, AlgMatElt, RngElt -> RngFunOrdIdl
ideal< O | T, S > : RngFunOrd, AlgMatElt, [RngFunOrdIdl] -> RngFunOrdIdl
x * O : RngElt, RngFunOrd -> RngFunOrdIdl
Ideal(P) : PlcFunElt -> RngFunOrdIdl
Ideals(D) : DivFunElt -> RngFunOrdIdl, RngFunOrdIdl
O !! I : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl
Arithmetic Operators
c / I : RngElt, RngFunOrdIdl -> RngFunOrdIdl
ColonIdeal(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
Roots of Ideals
IsPower(I, n) : RngFunOrdIdl, RngIntElt -> BoolElt, RngFunOrdIdl
Root(I, n) : RngFunOrdIdl, RngIntElt -> RngFunOrdIdl
IsSquare(I) : RngFunOrdIdl -> BoolElt, RngFunOrdIdl
SquareRoot(I) : RngFunOrdIdl -> RngFunOrdIdl
Example FldFunG_ideal-is-square (H55E30)
Predicates on Ideals
IsZero(I) : RngFunOrdIdl -> BoolElt
IsOne(I) : RngFunOrdIdl -> BoolElt
IsIntegral(I) : RngFunOrdIdl -> BoolElt
IsPrime(I) : RngFunOrdIdl -> BoolElt
IsPrincipal(I) : RngFunOrdIdl -> BoolElt, FldFunElt
Predicates on Prime Ideals
IsInert(P) : RngFunOrdIdl -> BoolElt
IsInert(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsRamified(P) : RngFunOrdIdl -> BoolElt
IsRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsSplit(P) : RngFunOrdIdl -> BoolElt
IsSplit(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsTamelyRamified(P) : RngFunOrdIdl -> BoolElt
IsTamelyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsTotallyRamified(P) : RngFunOrdIdl -> BoolElt
IsTotallyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsTotallySplit(P) : RngFunOrdIdl -> BoolElt
IsTotallySplit(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsUnramified(P) : RngFunOrdIdl -> BoolElt
IsUnramified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
IsWildlyRamified(P) : RngFunOrdIdl -> BoolElt
IsWildlyRamified(P, O) : RngFunOrdIdl, RngFunOrd -> BoolElt
Further Ideal Operations
I meet J : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
Gcd(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
Lcm(I, J) : RngFunOrdIdl, RngFunOrdIdl -> RngFunOrdIdl
Factorization(I) : RngFunOrdIdl -> [ <RngFunOrdIdl, RngIntElt> ]
Decomposition(O, p) : RngFunOrd, RngElt -> [ RngFunOrdIdl ]
Decomposition(O) : RngFunOrd -> [ RngFunOrdIdl ]
DecompositionType(O, p) : RngFunOrd, RngElt -> [ <RngIntElt, RngIntElt> ]
DecompositionType(O) : RngFunOrd -> [ <RngIntElt, RngIntElt> ]
MultiplicatorRing(I) : RngFunOrdIdl -> RngFunOrd
pMaximalOrder(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrd
pRadical(O, p) : RngFunOrd, RngFunOrdIdl -> RngFunOrdIdl
Valuation(a, P) : RngElt, RngFunOrdIdl -> RngIntElt
Order(I) : RngFunOrdIdl -> RngFunOrd
Denominator(I) : RngFunOrdIdl -> RngElt
Minimum(I) : RngFunOrdIdl -> Any
I meet R : RngFunOrdIdl, Rng -> Any
IntegralSplit(I) : RngFunOrdIdl -> RngFunOrdIdl, RngElt
Norm(I) : RngFunOrdIdl -> Any
TwoElement(I) : RngFunOrdIdl -> RngElt, RngElt
Generators(I) : RngFunOrdIdl -> [ RngFunOrdElt ]
Basis(I) : RngFunOrdIdl -> [FldFunElt]
BasisMatrix(I) : RngFunOrdIdl -> AlgMatElt
TransformationMatrix(I) : RngFunOrdIdl -> AlgMatElt, RngElt
Different(I) : RngFunOrdIdl -> RngFunOrdIdl
Divisor(I) : RngFunOrdIdl -> DivFunElt
Divisor(I, J) : RngFunOrdIdl, RngFunOrdIdl -> DivFunElt
Example FldFunG_ideals (H55E31)
Functions on Prime Ideals
RamificationIndex(I) : RngFunOrdIdl -> RngIntElt
Degree(I) : RngFunOrdIdl -> RngIntElt
ResidueClassField(I) : RngFunOrdIdl -> Rng, Map
Place(I) : RngFunOrdIdl -> PlcFunElt
Example FldFunG_order-ideals (H55E32)
Creation of Structures
Places(F) : FldFun -> PlcFun
General Function Field Places
Decomposition(F, P) : FldFun, PlcFunElt -> [ PlcFunElt ]
DecompositionType(F, P) : FldFun, PlcFunElt -> [ <RngIntElt, RngIntElt> ]
Zeros(a) : FldFunElt -> [ PlcFunElt ]
Poles(a) : FldFunElt -> [ PlcFunElt ]
S ! I : PlcFun, RngFunOrdIdl -> PlcFunElt
Support(D) : DivFunElt -> [ PlcFunElt ], [ RngIntElt ]
AssignNames(~P, s) : PlcFunElt, [ MonStgElt ] ->
Global Function Field Places
HasPlace(F, m) : FldFun, RngIntElt -> PlcFunElt
RandomPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
Example FldFunG_place-creation (H55E33)
Parent and Category
FunctionField(S) : PlcFun -> FldFun
DivisorGroup(F) : FldFun -> DivFun
General function fields
WeierstrassPlaces(F) : FldFunG -> [PlcFunElt]
Global Function Fields
NumberOfPlacesOfDegreeOneOverExactConstantField(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOverExactConstantField(F, m) : FldFun, RngIntElt -> RngIntElt
Arithmetic Operators
Quotrem(P, k) : PlcFunElt, RngIntElt -> DivFunElt, DivFunElt
Predicates on Elements
IsFinite(P) : PlcFunElt -> BoolElt
IsWeierstrassPlace(P) : PlcFunElt -> BoolElt
Other Element Operations
FunctionField(P) : PlcFunElt -> FldFun
Degree(P) : PlcFunElt -> RngIntElt
RamificationIndex(P) : PlcFunElt -> RngIntElt
InertiaDegree(P) : PlcFunElt -> RngIntElt
Minimum(P) : PlcFunElt -> RngElt
ResidueClassField(P) : PlcFunElt -> Rng
Evaluate(a, P) : RngElt, PlcFunElt -> RngElt
Lift(a, P) : RngElt, PlcFunElt -> FldFunElt
TwoGenerators(P) : PlcFunElt -> FldFunGElt, FldFunGElt
LocalUniformizer(P) : PlcFunElt -> FldFunGElt
Valuation(a, P) : FldFunElt, PlcFunElt -> RngIntElt
Ideal(P) : PlcFunElt -> RngFunOrdIdl
Norm(P) : PlcFunElt -> DivFunElt
Example FldFunG_places (H55E34)
Completion at Places
Completion(F, p) : FldFun, PlcFunElt -> RngSerLaur, Map
Creation of Structures
DivisorGroup(F) : FldFun -> DivFun
Creation of Elements
Divisor(P) : PlcFunElt -> DivFunElt
Div ! a : DivFun, RngElt -> DivFunElt
Div ! I : DivFun, RngFunOrdIdl -> DivFunElt
Divisor(I, J) : RngFunOrdIdl, RngFunOrdIdl -> DivFunElt
Identity(G) : DivFun -> DivFunElt
CanonicalDivisor(F) : FldFun -> DivFunElt
DifferentDivisor(F) : FldFun -> DivFunElt
AssignNames(~D, s) : DivFunElt, [ MonStgElt ] ->
Parent and Category
FunctionField(G) : DivFun -> FldFun
Places(F) : FldFun -> PlcFun
Structure Invariants
NumberOfSmoothDivisors(n, m, P) : RngIntElt, RngIntElt, SeqEnum[RngElt] -> RngElt
DivisorOfDegreeOne(F) : FldFun -> DivFunElt
Arithmetic Operators
Quotrem(D, k) : DivFunElt, RngIntElt -> DivFunElt, DivFunElt
GCD(D1, D2) : DivFunElt, DivFunElt -> DivFunElt
LCM(D1, D2) : DivFunElt, DivFunElt -> DivFunElt
Equality, Comparison and Membership
Predicates on Elements
IsCanonical(D) : DivFunElt -> BoolElt, DiffFunElt
Example FldFunG_divisors-simple_rel (H55E35)
Other Element Operations
FunctionField(D) : DivFunElt -> FldFun
Degree(D) : DivFunElt -> RngIntElt
Support(D) : DivFunElt -> [ PlcFunElt ]
Numerator(D) : DivFunElt -> DivFunElt
Denominator(D) : DivFunElt -> DivFunElt
Ideals(D) : DivFunElt -> RngFunOrdIdl, RngFunOrdIdl
Norm(D) : DivFunElt -> DivFunElt
Dimension(D) : DivFunElt -> RngIntElt
IndexOfSpeciality(D) : DivFunElt -> RngIntElt
ShortBasis(D : parameters) : DivFunElt -> [RngElt], [RngIntElt]
Basis(D : parameters) : DivFunElt -> [ FldFunElt ]
RiemannRochSpace(D) : DivFunElt -> ModFld, Map
Valuation(D, P) : DivFunElt, PlcFunElt -> RngIntElt
Reduction(D) : DivFunElt -> DivFunElt, RngIntElt, DivFunElt, FldFunElt
GapNumbers(D, P) : DivFunElt, PlcFunElt -> SeqEnum[RngIntElt]
GapNumbers(D) : DivFunElt -> SeqEnum[RngIntElt]
Example FldFunG_divisors (H55E36)
Example FldFunG_AlgReln1 (H55E37)
Example FldFunG_AlgReln2 (H55E38)
RamificationDivisor(D) : DivFunElt -> DivFunElt
WeierstrassPlaces(D) : DivFunElt -> [PlcFunElt]
IsWeierstrassPlace(D, P) : DivFunElt, PlcFunElt -> BoolElt
WronskianOrders(D) : DivFunElt -> [RngIntElt]
ComplementaryDivisor(D) : DivFunElt -> DivFunElt
DifferentialBasis(D) : DivFunElt -> [DiffFunElt]
DifferentialSpace(D) : DivFunElt -> ModFld, Map
Parametrization(F, D) : FldFun, DivFunElt -> FldFunElt, [FldFunRatUElt]
Functions related to Divisor Class Groups of Global Function Fields
ClassGroupGenerationBound(q, g) : RngIntElt, RngIntElt -> RngIntElt
ClassGroupGenerationBound(F) : FldFun -> RngIntElt
ClassNumberApproximation(F, e) : FldFun, FldPrElt -> FldReElt
ClassNumberApproximationBound(q, g, e) : RngIntElt, RngIntElt, -> RngIntElt
ClassGroup(F : parameters) : FldFun -> GrpAb, Map, Map
ClassGroupAbelianInvariants(F : parameters) : FldFun -> SeqEnum
ClassNumber(F) : FldFun -> RngIntElt
Example FldFunG_divisors-class (H55E39)
GlobalUnitGroup(F) : FldFun -> GrpAb, Map
IsGlobalUnit(a) : FldFunElt -> BoolElt
IsGlobalUnitWithPreimage(a) : FldFunElt -> BoolElt, GrpAbElt
PrincipalDivisorMap(F) : FldFun -> Map
ClassGroupExactSequence(F) : FldFun -> Map, Map, Map
SUnitGroup(S) : SetEnum[PlcFunElt] -> GrpAb, Map
IsSUnit(a, S) : FldFunElt, SetEnum[PlcFunElt] -> BoolElt
IsSUnitWithPreimage(a, S) : FldFunElt, SetEnum[PlcFunElt] -> BoolElt, GrpAbElt
SRegulator(S) : SetEnum[PlcFunElt] -> RngIntElt
SPrincipalDivisorMap(S) : SetEnum[PlcFunElt] -> Map
IsSPrincipal(D, S) : DivFunElt, SetEnum[PlcFunElt] -> BoolElt, FldFunElt
SClassGroup(S) : SetEnum[PlcFunElt] -> GrpAb, Map, Map
SClassGroupExactSequence(S) : SetEnum[PlcFunElt] -> Map, Map, Map
SClassGroupAbelianInvariants(S) : SetEnum[PlcFunElt] -> SeqEnum
SClassNumber(S) : SetEnum[PlcFunElt] -> RngIntElt
ClassGroupPRank(F) : FldFunG -> RngIntElt
HasseWittInvariant(F) : FldFunG -> RngIntElt
TateLichtenbaumPairing(D1, D2, m) : DivFunElt, DivFunElt, RngIntElt -> RngElt
Example FldFunG_tate (H55E40)
Creation of Structures
RayResidueRing(D) : DivFunElt -> GrpAb, Map
RayClassGroup(D) : DivFunElt -> GrpAb, Map
RayClassGroupDiscLog(y, D) : DivFunElt, DivFunElt -> GrpAbElt
Example FldFunG_classfield-structures (H55E41)
Creation of Class Fields
RayClassField(D, U) : DivFunElt, GrpAb -> [FldFun], [Map]
Example FldFunG_classfields (H55E42)
Properties of Class Fields
Conductor(m) : DivFunElt -> DivFunElt
Conductor(m, U) : DivFunElt, GrpAb -> DivFunElt
DiscriminantDivisor(m, U) : DivFunElt, GrpAb -> DivFunElt
DegreeOfExactConstantField(m) : DivFunElt -> RngIntElt
DegreeOfExactConstantField(m, U) : DivFunElt, GrpAb -> RngIntElt
Genus(m, U) : DivFunElt, GrpAb -> RngIntElt
DecompositionType(m, U, p) : DivFunElt, GrpAb, PlcFunElt -> [<f,e>]
NumberOfPlacesOfDegreeOne(m, U) : DivFunElt, GrpAb -> RngIntElt
The Ring of Finite Witt Vectors
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) : Rng -> [RngLocElt]
LocalRing(W) : RngWitt -> RngLoc, Map
ArtinSchreierMap(W) : RngWitt -> Map
ArtinSchreierImage(e) : RngWittElt -> RngWittElt
FunctionField(e) : RngWittElt -> FldFun, Map
Related Functions
StrongApproximation(m, S): DivFunElt, [<PlcFunElt, FldFunElt>] -> FldFunElt
Example FldFunG_strong-approximation (H55E43)
NonSpecialDivisor(m): DivFunElt -> DivFunElt, RngIntElt
NormGroup(F) : FldFun -> DivFunElt, GrpAb
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
Creation of Structures
DifferentialSpace(F) : FldFunG -> DiffFun
Creation of Elements
Differential(a) : FldFunGElt -> DiffFunElt
Identity(D) : DiffFun -> DiffFunElt
IsCanonical(D) : DivFunElt -> BoolElt, DiffFunElt
Related Structures
FunctionField(D) : DiffFun -> FldFun
FunctionField(d) : DiffFunElt -> FldFun
Subspaces
SpaceOfDifferentialsFirstKind(F) : FldFunG -> ModFld, Map
BasisOfDifferentialsFirstKind(F) : FldFunG -> SeqEnum[DiffFunElt]
DifferentialBasis(D) : DivFunElt -> [DiffFunElt]
DifferentialSpace(D) : DivFunElt -> ModFld, Map
Example FldFunG_div_diff (H55E44)
Structure Predicates
D1 eq D2 : DiffFun, DiffFun -> BoolElt
Arithmetic Operators
r * x : RngElt, DiffFunElt -> DiffFunElt
Equality and Membership
x eq y : DiffFunElt, DiffFunElt -> BoolElt
x in D : Any, DiffFun -> BoolElt
Predicates on Elements
IsExact(d) : DiffFunElt -> BoolElt, FldFunGElt
IsZero(d) : DiffFunElt -> BoolElt
Functions on Elements
Valuation(d, P) : DiffFunElt, PlcFunElt -> RngIntElt
Divisor(d) : DiffFunElt -> DivFunElt
Residue(d, P) : DiffFunElt, PlcFunElt -> RngElt
Example FldFunG_diff-fun (H55E45)
Module(L, R) : SeqEnum[ DiffFunElt ], Rng -> Mod, Map, SeqEnum[ ModElt ]
Relations(L, R) : SeqEnum[ DiffFunElt ], Rng -> ModTupRng
Example FldFunG_module-diff (H55E46)
Cartier(b) : DiffFunElt -> DiffFunElt
Other
CartierRepresentation(F) : FldFunG -> AlgMatElt, SeqEnum[DiffFunElt]
Example FldFunG_diff-cart (H55E47)
Weil Descent
WeilDescent(E,k) : FldFun, FldFin -> FldFunG, Map
ArtinSchreierExtension(c,a,b) : FldFin, FldFin, FldFin -> FldFun
WeilDescentDegree(E,k) : FldFun, FldFin -> RngIntElt
WeilDescentGenus(E,k) : FldFun, FldFin -> RngIntElt
MultiplyFrobenius(b,f,F) : RngElt, RngUPolElt, Map -> RngElt