Creation of Algebraic Function Fields and their Orders
Creation of Algebraic Function Fields
ext< K | f > : FldFunRat, RngUPolElt -> FldFun
FunctionField(f : parameters) : RngMPolElt -> FldFun
HermitianFunctionField(p, d) : RngIntElt, RngIntElt -> FldFun
AssignNames(~F, s) : FldFun, [ MonStgElt ] ->
Example FldFunG_Creation (H53E1)
Example FldFunG_creation-rel (H53E2)
Example FldFunG_creation_herm (H53E3)
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
MaximalOrder(O) : RngFunOrd -> RngFunOrd
ext<O | f> : RngFunOrd, RngUPolElt -> RngFunOrd
Example FldFunG_orders (H53E4)
Example FldFunG_int_cl (H53E5)
Other Related Structures
PrimeRing(F) : FldFun -> Rng
ConstantField(F) : FldFun -> Rng
ExactConstantField(F) : FldFunG -> Rng, Map
BaseRing(F) : Fld -> Rng
BaseRing(O) : RngFunOrd -> Rng
FieldOfFractions(O) : RngFunOrd -> FldFun
Places(F) : FldFun -> PlcFun
DivisorGroup(F) : FldFun -> DivFun
DifferentialSpace(F) : FldFun -> DiffFun
Example FldFunG_related-structures (H53E6)
WeilRestriction(E, n) : FldFun, RngIntElt -> FldFun, UserProgram
Curve(F) : FldFun -> Crv
HasCurve(F) : FldFun -> BoolElt
ProjectiveCurve(F) : FldFun -> Crv
Reduce(O) : RngFunOrd -> RngFunOrd
General Structure Invariants
Characteristic(F) : FldFun -> RngIntElt
Degree(F) : FldFun -> RngIntElt
AbsoluteDegree(F) : FldFun -> RngIntElt
DefiningPolynomial(F) : FldFun -> RngUPolElt
Basis(F) : FldFun -> SeqEnum[FldFunElt]
Discriminant(O) : RngFunOrd -> .
RationalExtensionRepresentation(F) : FldFunG -> FldFun
DimensionOfExactConstantField(F) : FldFun -> RngIntElt
Genus(F) : FldFun -> RngIntElt
Example FldFunG_invar (H53E7)
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]
Galois Groups
GaloisGroup(f) : RngUPolElt -> GrpPerm, [ FldPrElt, Any ]
Example FldFunG_GaloisGroups (H53E8)
Function Fields over the Rationals
Subfields(F) : FldFun -> SeqEnum[FldFun]
Example FldFunG_Subfields (H53E9)
Global Function Fields
NumberOfPlacesOfDegreeOne(F) : FldFun -> RngIntElt
NumberOfPlacesOfDegreeOne(F, m) : FldFun, RngIntElt -> RngIntElt
SerreBound(F) : FldFun -> RngIntElt
IharaBound(F) : FldFun -> RngIntElt
NumberOfPlacesOfDegreeOneBound(F) : FldFun -> RngIntElt
NumberOfPlaces(F, m) : FldFun, RngIntElt -> RngIntElt
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
Place(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
RandomPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
LPolynomial(F) : FldFun -> RngUPolElt
LPolynomial(F, m) : FldFun, RngIntElt -> RngUPolElt
ZetaFunction(F) : FldFun -> FldFunRatUElt
ZetaFunction(F, m) : FldFun, RngIntElt -> FldFunRatUElt
UnitRank(O) : RngFunOrd -> RngIntElt
UnitGroup(O) : RngFunOrd -> GrpAb, Map
Regulator(O) : RngFunOrd -> RngIntElt
PrincipalIdealMap(O) : RngFunOrd -> Map
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
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 (H53E10)
Structure Predicates
IsGlobal(F) : FldFun -> BoolElt
IsFiniteOrder(O) : RngFunOrd -> BoolElt
IsEquationOrder(O) : RngFunOrd -> BoolElt
IsMaximal(O) : RngFunOrd -> BoolElt
Homomorphisms
hom<F -> R | g> : FldFun, Rng, RngElt -> Map
Example FldFunG_hom (H53E11)
Creation of Elements
F . 1 : FldFun -> FldFunElt
Name(F, i) : FldFun, RngIntElt -> FldFunElt
F ! a : FldFun, . -> FldFunElt
O ! a : RngFunOrd, . -> RngFunOrdElt
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[FldFunRatUElt]
F ! [ a_0, a_1, ..., a_(n - 1) ] : FldFun, SeqEnum -> FldFunElt
O ! [ a_1, a_2, ..., a_(n) ] : RngFunOrd, SeqEnum -> RngFunOrdElt
Example FldFunG_Elements (H53E12)
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
Functions related to Orders and Integrality
IntegralSplit(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt, RngElt
Numerator(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt
Denominator(a, O) : FldFunElt, RngFunOrd -> 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
Divisor(a) : FldFunGElt -> DivFunElt
Zeros(a) : FldFunGElt -> SeqEnum[PlcFunElt]
Poles(a) : FldFunElt -> SeqEnum[PlcFunElt]
Degree(a) : FldFunElt -> RngIntElt
CommonZeros(L) : SeqEnum[ FldFunElt ] -> SeqEnum[ PlcFunElt ]
Example FldFunG_elements (H53E13)
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 (H53E14)
Other Operations on Elements
Expand(a, P) : FldFunGElt, PlcFunElt -> RngSerElt, FldFunGElt
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
Creation of Ideals
ideal< O | a_1, a_2, ... , a_m > : RngFunOrd, RngElt, ..., RngElt -> RngFunOrdIdl
x * O : RngElt, RngFunOrd -> RngFunOrdIdl
Ideal(P) : PlcFunElt -> RngFunOrdIdl
Ideals(D) : DivFunElt -> RngFunOrdIdl, RngFunOrdIdl
Arithmetic Operators
c / I : RngElt, RngFunOrdIdl -> RngFunOrdIdl
Predicates on Ideals
IsZero(I) : RngFunOrdIdl -> BoolElt
IsOne(I) : RngFunOrdIdl -> BoolElt
IsIntegral(I) : RngFunOrdIdl -> BoolElt
IsPrime(I) : RngFunOrdIdl -> BoolElt
IsPrincipal(I) : RngFunOrdIdl -> BoolElt, FldFunElt
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
Valuation(a, P) : RngElt, RngFunOrdIdl -> RngIntElt
Order(I) : RngFunOrdIdl -> RngFunOrd
Denominator(I) : RngFunOrdIdl -> RngElt
Minimum(I) : RngFunOrdIdl -> 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
RamificationIndex(I) : RngFunOrdIdl -> RngIntElt
Degree(I) : RngFunOrdIdl -> RngIntElt
ResidueClassField(I) : RngFunOrdIdl -> Rng, Map
Place(I) : RngFunOrdIdl -> PlcFunElt
Divisor(I) : RngFunOrdIdl -> DivFunElt
Divisor(I, J) : RngFunOrdIdl, RngFunOrdIdl -> DivFunElt
Example FldFunG_ideals (H53E15)
Example FldFunG_order-ideals (H53E16)
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
Place(F, m) : FldFun, RngIntElt -> PlcFunElt
RandomPlace(F, m) : FldFun, RngIntElt -> BoolElt, PlcFunElt
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
Parent and Category
FunctionField(S) : PlcFun -> FldFun
DivisorGroup(F) : FldFun -> DivFun
General function fields
WeierstrassPlaces(F) : FldFunG -> [PlcFunElt]
Global Function Fields
NumberOfPlacesOfDegreeOne(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOneBound(F, m) : FldFun, RngIntElt -> RngIntElt
NumberOfPlaces(F, m) : FldFun, RngIntElt -> RngIntElt
Places(F, m) : FldFun, RngIntElt -> SeqEnum[PlcFunElt]
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
Example FldFunG_places (H53E17)
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
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
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 (H53E18)
Example FldFunG_AlgReln1 (H53E19)
Example FldFunG_AlgReln2 (H53E20)
RamificationDivisor(D) : DivFunElt -> DivFunElt
WeierstrassPlaces(D) : DivFunElt -> [PlcFunElt]
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
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
Example FldFunG_global-function-fields (H53E21)
ClassGroupPRank(F) : FldFunG -> RngIntElt
HasseWittInvariant(F) : FldFunG -> RngIntElt
TateLichtenbaumPairing(D1, D2, m) : DivFunElt, DivFunElt, RngIntElt -> RngElt
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 (H53E22)
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
Module(L, R) : SeqEnum[ DiffFunElt ], Rng -> Mod, Map, SeqEnum[ ModElt ]
Relations(L, R) : SeqEnum[ DiffFunElt ], Rng -> ModTupRng
Cartier(b) : DiffFunElt -> DiffFunElt