- Introduction
- Creation of Algebraic Function Fields and their Orders
- Creation of Algebraic Function Fields
- Construction 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 (H45E5)
- Example FldFunG_int_cl (H45E6)
- 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
- O1 meet O2 : RngFunOrd, RngFunOrd -> RngFunOrd
- AsExtensionOf(O1, O2) : RngFunOrd, RngFunOrd -> RngFunOrd
- Example FldFunG_order-create-more (H45E7)
- Orders and Ideals
- Related Structures
- Parent and Category
- Other Related Structures
- PrimeRing(F) : FldFun -> Rng
- ConstantField(F) : FldFunG -> 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
- AbsoluteFunctionField(F) : FldFunG -> FldFunG
- UnderlyingRing(F) : FldFunG -> FldFunG
- Embed(F, L, a) : FldFun, FldFun, FldFunElt ->
- Places(F) : FldFunG -> PlcFun
- DivisorGroup(F) : FldFun -> DivFun
- DifferentialSpace(F) : FldFun -> DiffFun
- Example FldFunG_related-structures (H45E8)
- Example FldFunG_related-structures-rat-ext (H45E9)
- WeilRestriction(E, n) : FldFun, RngIntElt -> FldFun, UserProgram
- ConstantFieldExtension(F, E) : FldFun, Rng -> FldFun, Map
- Example FldFunG_cfe (H45E10)
- MonicModel(F) : FldFun -> FldFun
- Reduce(O) : RngFunOrd -> RngFunOrd
- Localization(O, p) : RngFunOrd, RngFunOrdIdl -> RngVal, Map
- General Structure Invariants
- Galois Groups
- Subfields
- Automorphism Group
- Automorphisms over the Base Field
- Automorphisms(K, k) : FldFun, FldFunG -> [Map]
- AutomorphismGroup(K, k) : FldFun, FldFunG -> GrpFP, Map
- Example FldFunG_Automorphisms (H45E19)
- IsSubfield(K, L) : FldFun, FldFun -> BoolElt, Map
- IsIsomorphicOverQt(K, L) : FldFun, FldFun -> BoolElt, Map
- Example FldFunG_IsSubfield (H45E20)
- 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
- Global Function Fields
- Functions relative to the Exact Constant Field
- Functions Relative to the Constant Field
- Places(F, m) : FldFunG, RngIntElt -> SeqEnum[PlcFunElt]
- HasPlace(F, m) : FldFunG, RngIntElt -> BoolElt, PlcFunElt
- HasRandomPlace(F, m) : FldFunG, RngIntElt -> BoolElt, PlcFunElt
- RandomPlace(F, m) : FldFunG, RngIntElt -> PlcFunElt
- Example FldFunG_global-function-fields (H45E22)
- Example FldFunG_global1 (H45E23)
- Functions related to Class Group
- Structure Predicates
- Homomorphisms
- hom<F -> R | g> : FldFun, Rng, RngElt -> Map
- hom< O -> R | g > : RngFunOrd, Rng, RngElt -> Map
- IsRingHomomorphism(m) : Map -> BoolElt
- Example FldFunG_hom (H45E27)
- hom< O -> R | b1, ..., bn > : RngFunOrd, Rng, RngElt, ..., RngElt -> Map
- Elements
- 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 | a0, a1, ..., an - 1> : FldFun, RngElt , ..., RngElt -> FldFunElt
- elt< O | a1, a2, ..., an> : RngFunOrd, RngElt , ..., RngElt -> RngFunOrdElt
- Random(F, m) : FldFunG, RngIntElt -> FldFunElt
- Parent and Category
- Sequence Conversions
- ElementToSequence(a) : FldFunElt -> SeqEnum[FldElt]
- Eltseq(a, R) : FldFunElt, FldFunG -> [FldFunGElt]
- Flat(a) : FldFunElt -> [FldFunGElt]
- F ! [ a0, a1, ..., an - 1 ] : FldFun, SeqEnum -> FldFunElt
- O ! [ a1, a2, ..., an ] : RngFunOrd, SeqEnum -> RngFunOrdElt
- Example FldFunG_Elements (H45E28)
- Arithmetic Operators
- Equality and Membership
- Predicates on Elements
- Functions related to Norm and Trace
- RepresentationMatrix(a) : FldFunGElt -> AlgMatElt
- Trace(a, R) : FldFunElt, Rng -> RngElt
- Norm(a, R) : FldFunElt, Rng -> RngElt
- CharacteristicPolynomial(a, R) : FldFunElt, Rng -> RngUPolElt
- MinimalPolynomial(a, R) : FldFunElt, Rng -> RngUPolElt
- AbsoluteMinimalPolynomial(a) : FldFunElt -> RngUPolElt
- RepresentationMatrix(a, R) : FldFunGElt, Rng -> AlgMatElt
- Example FldFunG_elements-norm-trace (H45E29)
- Functions related to Orders and Integrality
- IntegralSplit(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt, RngElt
- Numerator(a, O) : FldFunElt, RngFunOrd -> RngFunOrdElt
- Numerator(a) : FldFunOrdElt -> RngFunOrdElt
- Numerator(a, O) : FldFunOrdElt, RngFunOrd -> RngElt
- Denominator(a, O) : FldFunElt, RngFunOrd -> RngElt
- Denominator(a) : FldFunOrdElt -> RngElt
- Denominator(a, O) : FldFunOrdElt, RngFunOrd -> RngElt
- Min(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
- Development(a, P) : FldFunGElt, PlcFunElt -> RngSerElt
- Divisor(a) : FldFunGElt -> DivFunElt
- Zeros(a) : FldFunGElt -> [PlcFunElt]
- Zeros(F, a) : FldFunG, FldFunGElt -> [PlcFunElt]
- Poles(a) : FldFunGElt -> 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 (H45E30)
- 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 (H45E31)
- 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
- RationalReconstruction(e, f) : FldFunElt, RngUPolElt -> BoolElt, FldFunElt
- CoefficientHeight(a) : RngFunOrdElt -> RngIntElt
- CoefficientLength(a) : RngFunOrdElt -> RngIntElt
- Example FldFunG_elements-other_ops (H45E32)
- Ideals
- Creation of Ideals
- ideal< O | a1, a2, ... , am > : 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
- Parent and Category
- Arithmetic Operators
- Roots of Ideals
- Equality and Membership
- 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
- CoefficientIdeals(I) : RngFunOrdIdl -> [RngFunOrdIdl]
- Different(I) : RngFunOrdIdl -> RngFunOrdIdl
- Codifferent(I) : RngFunOrdIdl -> RngFunOrdIdl
- Divisor(I) : RngFunOrdIdl -> DivFunElt
- Divisor(I, J) : RngFunOrdIdl, RngFunOrdIdl -> DivFunElt
- CoefficientHeight(I) : RngFunOrdIdl -> RngIntElt
- CoefficientLength(I) : RngFunOrdIdl -> RngIntElt
- Example FldFunG_ideals (H45E34)
- Functions on Prime Ideals
- Quotient Rings
- Places
- Divisors
- Creation of Structures
- Creation of Elements
- Related Structures
- Structure Invariants
- Structure Predicates
- Element Operations
- Arithmetic Operators
- Equality, Comparison and Membership
- Predicates on Elements
- 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
- FiniteSplit(D) : DivFunElt -> 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 (H45E40)
- Example FldFunG_AlgReln1 (H45E41)
- Example FldFunG_AlgReln2 (H45E42)
- 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) : FldFunG -> RngIntElt
- ClassNumberApproximation(F, e) : FldFunG, FldReElt -> FldReElt
- ClassNumberApproximationBound(q, g, e) : RngIntElt, RngIntElt, FldReElt, -> RngIntElt
- ClassGroup(F : parameters) : FldFun -> GrpAb, Map, Map
- ClassGroupAbelianInvariants(F : parameters) : FldFun -> SeqEnum
- ClassNumber(F) : FldFun -> RngIntElt
- Example FldFunG_divisors-class (H45E43)
- GlobalUnitGroup(F) : FldFun -> GrpAb, Map
- IsGlobalUnit(a) : FldFunElt -> BoolElt
- IsGlobalUnitWithPreimage(a) : FldFunElt -> BoolElt, GrpAbElt
- PrincipalDivisorMap(F) : FldFunG -> Map
- ClassGroupExactSequence(F) : FldFunG -> 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 (H45E44)
- Differentials
- Creation of Structures
- Creation of Elements
- Related Structures
- Subspaces
- Structure Predicates
- Operations on Elements
- Arithmetic Operators
- Equality and Membership
- Predicates on Elements
- Functions on Elements
- Valuation(d, P) : DiffFunElt, PlcFunElt -> RngIntElt
- Divisor(d) : DiffFunElt -> DivFunElt
- Residue(d, P) : DiffFunElt, PlcFunElt -> RngElt
- Example FldFunG_diff-fun (H45E46)
- Module(L, R) : SeqEnum[ DiffFunElt ], Rng -> Mod, Map, SeqEnum[ ModElt ]
- Relations(L, R) : SeqEnum[ DiffFunElt ], Rng -> ModTupRng
- Example FldFunG_module-diff (H45E47)
- Cartier(b) : DiffFunElt -> DiffFunElt
- Other
- 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
- Example FldFunG_ghs-descent (H45E49)
- Function Field Database
- The Montes Algorithm
- Montes(f, p) : RngUPolElt, RngUPolElt -> SeqEnum, SeqEnum, RngIntElt
- Example FldFunG_montes-eg-1 (H45E51)
- Montes(K, p) : FldArith, RngElt ->
- Example FldFunG_montes-eg-2 (H45E52)
- SFL(P, s) : OMIdl, RngIntElt ->
- Example FldFunG_sfl (H45E53)
- SetUseMontes(f) : BoolElt ->
- GetUseMontes(t) : Cat -> BoolElt
- SetVerbose("Montes", v) : MonStgElt, RngIntElt ->
- Ideals in OM Representation
- Ideal Operations
- pIntegralBasis(I, p) : OMIdl, RngElt -> SeqEnum
- SIntegralBasis(I, S) : OMIdl, SeqEnum -> SeqEnum
- Basis(I) : OMIdl -> SeqEnum
- Example FldFunG_om-ideal-op (H45E56)
- TwoElement(I) : OMIdl -> FldArithElt, FldArithElt
- Norm(I) : OMIdl -> RngElt
- Valuation(alpha, P : parameters) : FldArithElt, OMIdl->RngIntElt,FldElt
- Valuation(I, P) : OMIdl, OMIdl -> RngIntElt
- a mod P : FldArithElt, OMIdl -> FldArithElt
- Factorization(I) : OMIdl -> SeqEnum
- Example FldFunG_om-ideal-ops (H45E57)
- ResidueField(I) : OMIdl -> Fld
- Degree(I) : OMIdl -> RngIntElt
- Example FldFunG_om-ideals-deg-res (H45E58)
- Divisors in OM representation
- Bibliography
V2.28, 13 July 2023