Algebraic Function Fields
- Introduction
- Creation of Algebraic Function Fields and their Orders
- Creation of Algebraic Function Fields
ext< K | f >: FldFunRat, RngUPolElt → FldFunext< K | f >: FldFun, RngUPolElt → FldFunFunctionField(f : parameters): RngUPolElt → FldFunFunctionField(f : parameters): RngMPolElt → FldFunFunctionField(S): [RngUPolElt] → FldFunHermitianFunctionField(p, d): RngIntElt, RngIntElt → FldFunHermitianFunctionField(q): RngIntElt → FldFunsub<F | S>: FldFun, [] → FldFunsub<F | s₁, ..., sᵣ >: FldFun, [] → FldFunAssignNames(~F, s): FldFun, [ MonStgElt ]AssignNames(~a, s): FldFunElt, [ MonStgElt ]FunctionField(R): Rng → FldFunG- Example: Creating a Function Field Extension
- Example: Creating a Relative Extension
- Example: Creating a Non-Simple Extension
- Example: Hermitian Function Field
- Construction of Orders of Algebraic Function Fields
EquationOrderFinite(F): FldFun → RngFunOrdMaximalOrderFinite(F): FldFun → RngFunOrdEquationOrderInfinite(F): FldFun → RngFunOrdMaximalOrderInfinite(F): FldFun → RngFunOrdIntegralClosure(R, F): Rng, FldFun → RngFunOrdEquationOrder(O): RngFunOrd → RngFunOrdMaximalOrder(O): RngFunOrd → RngFunOrdSetOrderMaximal(O, b): RngFunOrd, BoolEltext<O | f>: RngFunOrd, RngUPolElt → RngFunOrd- Example: Creating Orders of a Function Field
- Example: Integral Closure in a Function Field
Order(O, T, d): RngFunOrd, AlgMatElt, RngElt → RngFunOrdOrder(O, M): RngFunOrd, ModDed → RngFunOrdOrder(O, S): RngFunOrd, [FldFunElt] → RngFunOrdSimplify(O): RngFunOrd → RngFunOrdO1 + O2: RngFunOrd, RngFunOrd → RngFunOrdO1 meet O2: RngFunOrd, RngFunOrd → RngFunOrdAsExtensionOf(O1, O2): RngFunOrd, RngFunOrd → RngFunOrd- Example: Further Order Constructions
- Orders and Ideals
- Related Structures
- Parent and Category
- Other Related Structures
PrimeRing(F): FldFun → RngPrimeField(F): FldFun → RngPrimeRing(O): RngFunOrd → RngConstantField(F): FldFunG → RngDefiningConstantField(F): FldFunG → RngExactConstantField(F): FldFunG → Rng, MapBaseRing(F): FldFun → RngBaseField(F): FldFun → RngCoefficientRing(F): FldFun → RngCoefficientField(F): FldFun → RngISABaseField(F,G): Fld, Fld → BoolEltBaseRing(O): RngFunOrd → RngCoefficientRing(O): RngFunOrd → RngBaseRing(FF): FldFunOrd → RngBaseField(FF): FldFunOrd → RngCoefficientRing(FF): FldFunOrd → RngCoefficientField(FF): FldFunOrd → RngSubOrder(O): RngFunOrd → RngFunOrdFunctionField(O): RngFunOrd → FldFunFieldOfFractions(O): RngFunOrd → FldFunOrdFieldOfFractions(FF): FldFunOrd → FldFunOrdFieldOfFractions(F): FldFun → FldFunOrder(FF): FldFunOrd → RngFunOrdRationalExtensionRepresentation(F): FldFunG → FldFunAbsoluteOrder(O): RngFunOrd → RngFunOrdAbsoluteFunctionField(F): FldFunG → FldFunGUnderlyingRing(F): FldFunG → FldFunGUnderlyingField(F): FldFunG → FldFunGUnderlyingRing(F, R): FldFunG, Rng → FldFunGUnderlyingField(F, R): FldFunG, Rng → FldFunGEmbed(F, L, a): FldFun, FldFun, FldFunEltEmbed(F, L, s): FldFun, FldFun, [FldFunElt]HasEmbedding(O1, O2): RngFunOrd, RngFunOrd → BoolCanComputeEmbedding(O1, O2): RngFunOrd, RngFunOrd → BoolPlaces(F): FldFunG → PlcFunDivisorGroup(F): FldFun → DivFunDifferentialSpace(F): FldFun → DiffFun- Example: Constant and Coefficient Fields
- Example: Underlying Rings of a Relative Function Field
WeilRestriction(E, n): FldFun, RngIntElt → FldFun, UserProgramConstantFieldExtension(F, E): FldFun, Rng → FldFun, Map- Example: Extending to the Exact Constant Field
MonicModel(F): FldFun → FldFunReduce(O): RngFunOrd → RngFunOrdLocalization(O, p): RngFunOrd, RngFunOrdIdl → RngVal, Map
- General Structure Invariants
Characteristic(F): FldFun → RngIntEltCharacteristic(O): RngFunOrd → RngIntEltIsPerfect(F): Fld → BoolEltDegree(F): FldFunG → RngIntEltDegree(F, G): FldFun, FldFun → RngIntEltDegree(O): RngFunOrd → RngIntEltAbsoluteDegree(F): FldFunG → RngIntEltAbsoluteDegree(O): RngFunOrd → RngIntEltDefiningPolynomial(F): FldFun → RngUPolEltDefiningPolynomial(O): RngFunOrd → RngUPolEltDefiningPolynomials(F): FldFun → [RngUPolElt]DefiningPolynomials(O): RngFunOrd → [RngUPolElt]Basis(F): FldFunG → SeqEnum[FldFunElt]Basis(O): RngFunOrd → SeqEnum[FldFunElt]Basis(O, R): RngFunOrd, Rng → SeqEnum[RngElt]TransformationMatrix(O1, O2): RngFunOrd, RngFunOrd → AlgMatElt, RngEltCoefficientIdeals(O): RngFunOrd → [RngFunOrdIdl]BasisMatrix(O): RngFunOrd → AlgMatEltPrimitiveElement(O): RngFunOrd → RngFunOrdEltDiscriminant(O): RngFunOrd → .AbsoluteDiscriminant(O): RngFunOrd → .DimensionOfExactConstantField(F): FldFunG → RngIntEltDegreeOfExactConstantField(F): FldFunG → RngIntEltGenus(F): FldFunG → RngIntElt- Example: Invariants of a Relative Function Field
- Example: Invariants of a Non-Simple Field
GapNumbers(F): FldFunG → SeqEnum[RngIntElt]GapNumbers(F, P): FldFunG, PlcFunElt → SeqEnum[RngIntElt]GapNumbers(P): PlcFunElt → SeqEnum[RngIntElt]SeparatingElement(F): FldFunG → FldFunGEltRamificationDivisor(F): FldFunG → DivFunEltWeierstrassPlaces(F): FldFunG → [PlcFunElt]WronskianOrders(F): FldFunG → [RngIntElt]Different(O): RngFunOrd → RngFunOrdIdlIndex(O, S): RngFunOrd, RngFunOrd → Any
- Galois Groups
GaloisGroup(f): RngUPolElt → GrpPerm, [ RngElt ], GaloisDataGaloisGroup(F): FldFun → GrpPerm, [RngElt], GaloisDataGaloisGroup(F): FldFun[FldFun[FldFunRat]] → GrpPerm, [RngElt], GaloisDataGaloisGroup(F): FldFun[FldFin] → GrpPerm, [RngElt], GaloisDataGaloisGroup(F): FldFun[FldFunRat] → GrpPerm, [RngElt], GaloisDataGaloisGroup(F): FldFun[FldRat] → GrpPerm, [RngElt], GaloisData- Example: Galois Group over a Rational Function Field
- Example: Galois Groups in Positive Characteristic
GeometricGaloisGroup(f): RngUPolElt → GrpPerm, RngUPolElt, GaloisDataHilbertIrreducibilityCurves(f): RngUPolElt → SetEnum, SeqEnum- Example: Geometric Galois Group
- Splitting Fields
- Subfields
- Automorphism Group
- Automorphisms over the Base Field
Automorphisms(K, k): FldFun, FldFunG → [Map]AutomorphismGroup(K, k): FldFun, FldFunG → GrpFP, Map- Example: Automorphisms of a Degree 7 Extension
IsSubfield(K, L): FldFun, FldFun → BoolElt, MapIsIsomorphicOverQt(K, L): FldFun, FldFun → BoolElt, Map- Example: Subfields and Isomorphism Testing
- General Automorphisms
Isomorphisms(K, E): FldFunG, FldFunG → [Map]IsIsomorphic(K, E): FldFunG, FldFunG → BoolElt, MapAutomorphisms(K): FldFunG → [Map]Isomorphisms(K,E,p1,p2): FldFunG, FldFunG, PlcFunElt, PlcFunElt → [Map]Automorphisms(K,p1,p2): FldFunG, PlcFunElt, PlcFunElt → [Map]AutomorphismGroup(K): FldFunG → GrpFP, MapAutomorphismGroup(K,f): FldFunG, Map → Grp, Map, SeqEnum
- Field Morphisms
- Global Function Fields
- Functions relative to the Exact Constant Field
NumberOfPlacesOfDegreeOverExactConstantField(F, m): FldFun, RngIntElt → RngIntEltNumberOfPlacesDegECF(F, m): FldFun, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantField(F): FldFunG → RngIntEltNumberOfPlacesOfDegreeOneECF(F): FldFunG → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantField(F, m): FldFunG, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneECF(F, m): FldFun, RngIntElt → RngIntEltSerreBound(F): FldFunG → RngIntEltSerreBound(F, m): FldFunG, RngIntElt → RngIntEltSerreBound(q, g): RngIntElt, RngIntElt → RngIntEltIharaBound(F): FldFunG → RngIntEltIharaBound(F, m): FldFunG, RngIntElt → RngIntEltIharaBound(q, g): RngIntElt, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneECFBound(F): FldFunG → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F): FldFunG → RngIntEltNumberOfPlacesOfDegreeOneECFBound(F, m): FldFun, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F, m): FldFunG, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneECFBound(q, g): RngIntElt, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantFieldBound(q, g): RngIntElt, RngIntElt → RngIntEltLPolynomial(F): FldFunG → RngUPolEltLPolynomial(F, m): FldFunG, RngIntElt → RngUPolEltZetaFunction(F): FldFunG → FldFunRatUEltZetaFunction(F, m): FldFunG, RngIntElt → FldFunRatUElt
- Functions Relative to the Constant Field
Places(F, m): FldFunG, RngIntElt → SeqEnum[PlcFunElt]HasPlace(F, m): FldFunG, RngIntElt → BoolElt, PlcFunEltHasRandomPlace(F, m): FldFunG, RngIntElt → BoolElt, PlcFunEltRandomPlace(F, m): FldFunG, RngIntElt → PlcFunElt- Example: Invariants of a Global Function Field
- Example: Invariants of a Global Relative Field
- Functions related to Class Group
- Structure Predicates
- Homomorphisms
hom<F -> R | g>: FldFun, Rng, RngElt → Maphom<F -> R | cf, g>: FldFun, Rng, Map, RngElt → Maphom< O -> R | g >: RngFunOrd, Rng, RngElt → Maphom< O -> R | cf, g >: RngFunOrd, Rng, Map, RngElt → MapIsRingHomomorphism(m): Map → BoolElt- Example: Homomorphisms of Relative Function Fields
hom< O -> R | b₁, ..., bₙ >: RngFunOrd, Rng, RngElt, ..., RngElt → Maphom< O -> R | m, b₁, ..., bₙ >: RngFunOrd, Rng, Map, RngElt, ..., RngElt → Map
- Elements
- Creation of Elements
F . 1: FldFun → FldFunEltF . 2: FldFun → FldFunEltName(F, i): FldFun, RngIntElt → FldFunEltO . i: RngFunOrd, RngIntElt → FldFunOrdEltFF . i: FldFunOrd, RngIntElt → FldFunOrdEltF ! a: FldFun, . → FldFunEltelt<F | a>: FldFun, . → FldFunEltO ! a: RngFunOrd, . → RngFunOrdEltelt<O | a>: RngFunOrd, . → RngFunOrdEltFF ! a: FldFunOrd, Any → FldFunOrdEltelt<FF | a>: FldFunOrd, Any → FldFunOrdEltelt< F | a₀, a₁, ..., aₙ₋₁>: FldFun, RngElt, ..., RngElt → FldFunEltelt< O | a₁, a₂, ..., aₙ>: RngFunOrd, RngElt, ..., RngElt → RngFunOrdEltelt< FF | a₁, a₂, ..., aₙ>: FldFunOrd, RngElt, ..., RngElt → FldFunOrdEltOne(F): FldFun → FldFunEltOne(O): RngFunOrd → RngFunOrdEltIdentity(F): FldFun → FldFunEltIdentity(O): RngFunOrd → RngFunOrdEltZero(F): FldFun → FldFunEltZero(O): RngFunOrd → RngFunOrdEltRepresentative(F): FldFun → FldFunEltRepresentative(O): RngFunOrd → RngFunOrdEltRandom(F, m): FldFunG, RngIntElt → FldFunEltRandom(O, m): RngFunOrd, RngIntElt → RngFunOrdElt
- Parent and Category
- Sequence Conversions
ElementToSequence(a): FldFunElt → SeqEnum[FldElt]ElementToSequence(a): RngFunOrdElt → SeqEnum[RngElt]ElementToSequence(a): FldFunOrdElt → SeqEnum[RngElt]Eltseq(a): FldFunElt → SeqEnum[FldElt]Eltseq(a): RngFunOrdElt → SeqEnum[RngElt]Eltseq(a): FldFunOrdElt → SeqEnum[RngElt]Eltseq(a, R): FldFunElt, FldFunG → [FldFunGElt]Flat(a): FldFunElt → [FldFunGElt]F ! [ a₀, a₁, ..., aₙ₋₁ ]: FldFun, SeqEnum → FldFunEltO ! [ a₁, a₂, ..., aₙ ]: RngFunOrd, SeqEnum → RngFunOrdElt- Example: Creating Function Field Elements
- Arithmetic Operators
+ a: FldFunElt → FldFunElt+ a: RngFunOrdElt → RngFunOrdElt+ a: FldFunOrdElt → FldFunOrdElt- a: FldFunElt → FldFunElt- a: RngFunOrdElt → RngFunOrdElt- a: FldFunOrdElt → FldFunOrdElta + b: FldFunElt, FldFunElt → FldFunElta + b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElta + b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElta - b: FldFunElt, FldFunElt → FldFunElta - b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElta - b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElta * b: FldFunElt, FldFunElt → FldFunElta * b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElta * b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElta div b: FldFunElt, FldFunElt → FldFunElta div b: FldFunElt, RngElt → FldFunElta div b: RngFunOrdElt, RngFunOrdElt → RngFunOrdElta div b: RngFunOrdElt, RngElt → RngFunOrdElta / b: FldFunElt, FldFunElt → FldFunElta / b: RngFunOrdElt, RngFunOrdElt → FldFunGElta / b: FldFunOrdElt, FldFunOrdElt → FldFunOrdElta ^ k: FldFunElt, RngIntElt → FldFunElta ^ k: RngFunOrdElt, RngIntElt → RngFunOrdElta ^ k: FldFunOrdElt, RngIntElt → FldFunOrdEltModexp(a, k, m): RngFunOrdElt, RngIntElt, RngUPolElt → RngFunOrdEltModexp(a, k, m): RngFunOrdElt, RngIntElt, RngValElt → RngFunOrdElta mod I: RngFunOrdElt, RngFunOrdIdl → RngFunOrdEltModinv(a, m): RngFunOrdElt, RngFunOrdIdl → RngFunOrdEltModinv(a, m): RngFunOrdElt, RngUPolElt → RngFunOrdEltModinv(a, m): RngFunOrdElt, RngValElt → RngFunOrdElt
- Equality and Membership
a eq b: FldFunElt, FldFunElt → BoolElta eq b: RngFunOrdElt, RngFunOrdElt → BoolElta eq b: FldFunOrdElt, FldFunOrdElt → BoolElta ne b: FldFunElt, FldFunElt → BoolElta ne b: RngFunOrdElt, RngFunOrdElt → BoolElta ne b: FldFunOrdElt, FldFunOrdElt → BoolElta in F: RngElt, FldFun → BoolElta in O: RngElt, RngFunOrd → BoolElta in FF: RngElt, FldFunOrd → BoolElta notin F: RngElt, FldFun → BoolElta notin O: RngElt, RngFunOrd → BoolElta notin FF: RngElt, FldFunOrd → BoolElt
- Predicates on Elements
IsDivisibleBy(a, b): FldFunElt, FldFunElt → BoolElt, FldFunEltIsDivisibleBy(a, b): RngFunOrdElt, RngFunOrdElt → BoolElt, RngFunOrdEltIsZero(a): FldFunElt → BoolEltIsZero(a): RngFunOrdElt → BoolEltIsZero(a): FldFunOrdElt → BoolEltIsOne(a): FldFunElt → BoolEltIsOne(a): RngFunOrdElt → BoolEltIsOne(a): FldFunOrdElt → BoolEltIsMinusOne(a): FldFunElt → BoolEltIsMinusOne(a): RngFunOrdElt → BoolEltIsMinusOne(a): FldFunOrdElt → BoolEltIsNilpotent(a): FldFunElt → BoolEltIsNilpotent(a): RngFunOrdElt → BoolEltIsIdempotent(a): FldFunElt → BoolEltIsIdempotent(a): RngFunOrdElt → BoolEltIsUnit(a): FldFunElt → BoolEltIsUnit(a): RngFunOrdElt → BoolEltIsUnit(a): FldFunOrdElt → BoolEltIsZeroDivisor(a): FldFunElt → BoolEltIsZeroDivisor(a): RngFunOrdElt → BoolEltIsRegular(a): FldFunElt → BoolEltIsRegular(a): RngFunOrdElt → BoolEltIsIrreducible(a): FldFunElt → BoolEltIsIrreducible(a): RngFunOrdElt → BoolEltIsPrime(a): FldFunElt → BoolEltIsPrime(a): RngFunOrdElt → BoolEltIsSeparating(a): FldFunGElt → BoolEltIsConstant(a): FldFunGElt → BoolElt, RngEltIsConstant(a): RngFunOrdElt → BoolElt, RngEltIsGlobalUnit(a): FldFunElt → BoolEltIsGlobalUnitWithPreimage(a): FldFunElt → BoolElt, GrpAbEltIsUnitWithPreimage(a): RngFunOrdElt → BoolElt, GrpAbElt
- Functions related to Norm and Trace
Trace(a): FldFunGElt → FldFunGEltTrace(a): RngFunOrdElt → RngEltNorm(a): FldFunGElt → FldFunGEltNorm(a): RngFunOrdElt → RngEltMinimalPolynomial(a): FldFunElt → RngUPolEltMinimalPolynomial(a): RngFunOrdElt → RngUPolEltMinimalPolynomial(a): FldFunOrdElt → RngUPolEltCharacteristicPolynomial(a): FldFunElt → RngUPolEltCharacteristicPolynomial(a): RngFunOrdElt → RngUPolEltRepresentationMatrix(a): FldFunGElt → AlgMatEltRepresentationMatrix(a): RngFunOrdElt → AlgMatEltTrace(a, R): FldFunElt, Rng → RngEltTrace(a, R): RngFunOrdElt, Rng → RngEltNorm(a, R): FldFunElt, Rng → RngEltNorm(a, R): RngFunOrdElt, Rng → RngEltCharacteristicPolynomial(a, R): FldFunElt, Rng → RngUPolEltCharacteristicPolynomial(a, R): RngFunOrdElt, Rng → RngUPolEltMinimalPolynomial(a, R): FldFunElt, Rng → RngUPolEltMinimalPolynomial(a, R): RngFunOrdElt, Rng → RngUPolEltAbsoluteMinimalPolynomial(a): FldFunElt → RngUPolEltRepresentationMatrix(a, R): FldFunGElt, Rng → AlgMatEltRepresentationMatrix(a, R): RngFunOrdElt, Rng → AlgMatElt- Example: Norm and Trace of Elements
- Functions related to Orders and Integrality
IntegralSplit(a, O): FldFunElt, RngFunOrd → RngFunOrdElt, RngEltIntegralSplit(a, O): RngFunOrdElt, RngFunOrd → RngFunOrdElt, RngEltIntegralSplit(a, O): FldFunOrdElt, RngFunOrd → RngFunOrdElt, RngEltNumerator(a, O): FldFunElt, RngFunOrd → RngFunOrdEltNumerator(a): FldFunOrdElt → RngFunOrdEltNumerator(a, O): FldFunOrdElt, RngFunOrd → RngEltDenominator(a, O): FldFunElt, RngFunOrd → RngEltDenominator(a): FldFunOrdElt → RngEltDenominator(a, O): FldFunOrdElt, RngFunOrd → RngEltMin(a, O): FldFunElt, RngFunOrd → RngElt, RngEltMinimum(a, O): FldFunElt, RngFunOrd → RngElt, RngEltMinimum(a, O): RngFunOrdElt, RngFunOrd → RngElt, RngEltMin(a, O): RngFunOrdElt, RngFunOrd → RngElt, RngElt
- Functions related to Places and Divisors
Evaluate(a, P): FldFunElt, PlcFunElt → RngEltEvaluate(a, P): RngFunOrdElt, PlcFunElt → RngEltLift(a, P): RngElt, PlcFunElt → FldFunEltValuation(a, P): FldFunElt, PlcFunElt → RngIntEltValuation(a, P): RngFunOrdElt, PlcFunElt → RngIntEltExpand(a, P): FldFunGElt, PlcFunElt → RngSerElt, FldFunGEltDevelopment(a, P): FldFunGElt, PlcFunElt → RngSerEltDivisor(a): FldFunGElt → DivFunEltDivisor(a): RngFunOrdElt → DivFunEltPrincipalDivisor(a): FldFunGElt → DivFunEltPrincipalDivisor(a): RngFunOrdElt → DivFunEltZeros(a): FldFunGElt → [PlcFunElt]Zeros(a): RngFunOrdElt → [PlcFunElt]Zeroes(a): FldFunGElt → [PlcFunElt]Zeroes(a): RngFunOrdElt → [PlcFunElt]Zeros(F, a): FldFunG, FldFunGElt → [PlcFunElt]Zeroes(F, a): FldFunG, FldFunGElt → [PlcFunElt]Poles(a): FldFunGElt → SeqEnum[PlcFunElt]Poles(a): RngFunOrdElt → SeqEnum[PlcFunElt]Poles(F, a): FldFun, FldFunGElt → [PlcFunElt]Degree(a): FldFunElt → RngIntEltDegree(a): RngFunOrdElt → RngIntEltCommonZeros(L): [FldFunGElt] → [PlcFunElt]CommonZeros(F, L): FldFunG, SeqEnum[ FldFunGElt ] → SeqEnum[ PlcFunElt ]- Example: Minimal Polynomial and Representation Matrix
Module(L, R): SeqEnum[ FldFunGElt ], Rng → Mod, Map, SeqEnum[ ModElt ]Relations(L, R): SeqEnum[ FldFunElt ], Rng → ModTupRngRelations(L, R, m): SeqEnum[ FldFunElt ], Rng, RngIntElt → ModTupRngRoots(f, D): RngUPolElt, DivFunElt → SeqEnum[ FldFunElt ]- Example: Module and Relations over Relative Fields
- Other Operations on Elements
ProductRepresentation(a): FldFunGElt → [FldFunGElt], [RngIntElt]ProductRepresentation(a): RngFunOrdElt → [RngElt], [RngIntElt]ProductRepresentation(Q, S): [FldFunGElt], [RngIntElt] → FldFunGEltPowerProduct(Q, S): [FldFunGElt], [RngIntElt] → FldFunGEltRationalFunction(a): FldFunGElt → RngEltRationalFunction(a, R): FldFunGElt, Rng → RngEltDifferentiation(x, a): FldFunGElt, FldFunGElt → FldFunGEltDifferentiation(x, n, a): FldFunGElt, RngIntElt, FldFunGElt → FldFunGEltDifferentiationSequence(x, n, a): FldFunGElt, RngIntElt, FldFunGElt → SeqEnumPrimePowerRepresentation(x, k, a): FldFunGElt, RngIntElt, FldFunGElt → SeqEnumDifferent(a): RngFunOrdElt → RngFunOrdEltRationalReconstruction(e, f): FldFunElt, RngUPolElt → BoolElt, FldFunEltCoefficientHeight(a): RngFunOrdElt → RngIntEltCoefficientHeight(a): FldFunElt → RngIntEltCoefficientLength(a): RngFunOrdElt → RngIntEltCoefficientLength(a): FldFunElt → RngIntElt- Example: Product Representation of Elements
- Ideals
- Creation of Ideals
ideal< O | a₁, a₂, ... , aₘ >: RngFunOrd, RngElt, ..., RngElt → RngFunOrdIdlideal< O | T, d >: RngFunOrd, AlgMatElt, RngElt → RngFunOrdIdlideal< O | T, d >: RngFunOrd, ModDed, RngElt → RngFunOrdIdlideal< O | T, S >: RngFunOrd, AlgMatElt, [RngFunOrdIdl] → RngFunOrdIdlideal< O | T, I₁, ..., Iₙ >: RngFunOrd, AlgMatElt, RngFunOrdIdl, ..., RngFunOrdIdl → RngFunOrdIdlx * O: RngElt, RngFunOrd → RngFunOrdIdlO * x: RngFunOrd, RngElt → RngFunOrdIdlIdeal(P): PlcFunElt → RngFunOrdIdlIdeals(D): DivFunElt → RngFunOrdIdl, RngFunOrdIdlO !! I: RngFunOrd, RngFunOrdIdl → RngFunOrdIdl
- Parent and Category
- Arithmetic Operators
I + J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlI * J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlI / J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlI ^ k: RngFunOrdIdl, RngIntElt → RngFunOrdIdlc * I: RngElt, RngFunOrdIdl → RngFunOrdIdlI * c: RngFunOrdIdl, RngElt → RngFunOrdIdlI / c: RngFunOrdIdl, RngElt → RngFunOrdIdlc / I: RngElt, RngFunOrdIdl → RngFunOrdIdlIdealQuotient(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlColonIdeal(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlChineseRemainderTheorem(I1, I2, e1, e2): RngFunOrdIdl, RngFunOrdIdl, RngFunOrdElt, RngFunOrdElt → RngFunOrdEltCRT(I1, I2, e1, e2): RngFunOrdIdl, RngFunOrdIdl, RngFunOrdElt, RngFunOrdElt → RngOrdElt
- Roots of Ideals
- Equality and Membership
- Predicates on Ideals
IsZero(I): RngFunOrdIdl → BoolEltIsOne(I): RngFunOrdIdl → BoolEltIsIntegral(I): RngFunOrdIdl → BoolEltIsPrime(I): RngFunOrdIdl → BoolEltIsPrincipal(I): RngFunOrdIdl → BoolElt, FldFunElt- Predicates on Prime Ideals
IsInert(P): RngFunOrdIdl → BoolEltIsInert(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsInert(P, O): RngElt, RngFunOrd → BoolEltIsRamified(P): RngFunOrdIdl → BoolEltIsRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsRamified(P, O): RngElt, RngFunOrd → BoolEltIsSplit(P): RngFunOrdIdl → BoolEltIsSplit(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsSplit(P, O): RngElt, RngFunOrd → BoolEltIsTamelyRamified(P): RngFunOrdIdl → BoolEltIsTamelyRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsTamelyRamified(P, O): RngElt, RngFunOrd → BoolEltIsTotallyRamified(P): RngFunOrdIdl → BoolEltIsTotallyRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsTotallyRamified(P, O): RngElt, RngFunOrd → BoolEltIsTotallySplit(P): RngFunOrdIdl → BoolEltIsTotallySplit(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsTotallySplit(P, O): RngElt, RngFunOrd → BoolEltIsUnramified(P): RngFunOrdIdl → BoolEltIsUnramified(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsUnramified(P, O): RngElt, RngFunOrd → BoolEltIsWildlyRamified(P): RngFunOrdIdl → BoolEltIsWildlyRamified(P, O): RngFunOrdIdl, RngFunOrd → BoolEltIsWildlyRamified(P, O): RngElt, RngFunOrd → BoolElt
- Further Ideal Operations
I meet J: RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlGcd(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlLcm(I, J): RngFunOrdIdl, RngFunOrdIdl → RngFunOrdIdlFactorization(I): RngFunOrdIdl → [ <RngFunOrdIdl, RngIntElt> ]Factorisation(I): RngFunOrdIdl → [<RngFunOrdIdl, RngIntElt>]Decomposition(O, p): RngFunOrd, RngElt → [ RngFunOrdIdl ]Decomposition(O, p): RngFunOrd, RngFunOrdIdl → [ RngFunOrdIdl ]Decomposition(O): RngFunOrd → [ RngFunOrdIdl ]DecompositionType(O, p): RngFunOrd, RngElt → [ <RngIntElt, RngIntElt> ]DecompositionType(O, p): RngFunOrd, RngFunOrdIdl → [ <RngIntElt, RngIntElt> ]DecompositionType(O): RngFunOrd → [ <RngIntElt, RngIntElt> ]MultiplicatorRing(I): RngFunOrdIdl → RngFunOrdpMaximalOrder(O, p): RngFunOrd, RngFunOrdIdl → RngFunOrdpRadical(O, p): RngFunOrd, RngFunOrdIdl → RngFunOrdIdlValuation(a, P): RngElt, RngFunOrdIdl → RngIntEltValuation(I, P): RngFunOrdIdl, RngFunOrdIdl → RngIntEltOrder(I): RngFunOrdIdl → RngFunOrdDenominator(I): RngFunOrdIdl → RngEltMinimum(I): RngFunOrdIdl → AnyI meet R: RngFunOrdIdl, Rng → AnyIntegralSplit(I): RngFunOrdIdl → RngFunOrdIdl, RngEltNorm(I): RngFunOrdIdl → AnyTwoElement(I): RngFunOrdIdl → RngElt, RngEltGenerators(I): RngFunOrdIdl → [ RngFunOrdElt ]Basis(I): RngFunOrdIdl → [FldFunElt]Basis(I, R): RngFunOrdIdl, Rng → [ RngElt ]BasisMatrix(I): RngFunOrdIdl → AlgMatEltTransformationMatrix(I): RngFunOrdIdl → AlgMatElt, RngEltCoefficientIdeals(I): RngFunOrdIdl → [RngFunOrdIdl]Different(I): RngFunOrdIdl → RngFunOrdIdlCodifferent(I): RngFunOrdIdl → RngFunOrdIdlDivisor(I): RngFunOrdIdl → DivFunEltDivisor(I, J): RngFunOrdIdl, RngFunOrdIdl → DivFunEltCoefficientHeight(I): RngFunOrdIdl → RngIntEltCoefficientLength(I): RngFunOrdIdl → RngIntElt- Example: Ideal Operations in a Relative Field
- Functions on Prime Ideals
- Quotient Rings
- Operations on Quotient Rings
quo< O | I >: RngFunOrd, RngFunOrdIdl → RngFunOrdResquo< O | M >: RngFunOrd, ModDed → RngFunOrdResquo< O | M >: RngFunOrd, AlgMatElt → RngFunOrdResquo< O | S >: RngFunOrd, RngElt, ..., RngElt → RngFunOrdResquo< O | p >: RngFunOrd, RngUPolElt → RngFunOrdResquo< O | p, r >: RngFunOrd, RngUPolElt, RngIntElt → RngFunOrdResModulus(OQ): RngFunOrdRes → RngFunOrdIdl- Example: Quotient Rings of Ideals
- Elements of Quotients
- Places
- Creation of Structures
- Creation of Elements
- General Function Field Places
Decomposition(F, P): FldFunG, PlcFunElt → [ PlcFunElt ]DecompositionType(F, P): FldFun, PlcFunElt → [ <RngIntElt, RngIntElt> ]Zeros(a): FldFunElt → [ PlcFunElt ]Zeros(a): RngFunOrdElt → [ PlcFunElt ]Poles(a): FldFunElt → [ PlcFunElt ]Poles(a): RngFunOrdElt → [ PlcFunElt ]S ! I: PlcFun, RngFunOrdIdl → PlcFunEltPlace(I): RngFunOrdIdl → PlcFunEltSupport(D): DivFunElt → [ PlcFunElt ], [ RngIntElt ]Support(P): PlcFunElt → [ PlcFunElt ], [ RngIntElt ]AssignNames(~P, s): PlcFunElt, [ MonStgElt ]InfinitePlaces(F): FldFun → [PlcFunElt]
- Global Function Field Places
- Related Structures
- Structure Invariants
- General Function Fields
- Global Function Fields
NumberOfPlacesOfDegreeOneOverExactConstantField(F, m): FldFun, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneECF(F, m): FldFunG, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantFieldBound(F, m): FldFun, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneECFBound(F, m): FldFunG, RngIntElt → RngIntEltNumberOfPlacesOfDegreeOverExactConstantField(F, m): FldFunG, RngIntElt → RngIntEltNumberOfPlacesDegECF(F, m): FldFunG, RngIntElt → RngIntElt
- Structure Predicates
- Element Operations
- Parent and Category
- Arithmetic Operators
- Equality and Membership
- Predicates on Elements
- Other Element Operations
FunctionField(P): PlcFunElt → FldFunDegree(P): PlcFunElt → RngIntEltRamificationIndex(P): PlcFunElt → RngIntEltRamificationDegree(P): PlcFunElt → RngIntEltInertiaDegree(P): PlcFunElt → RngIntEltResidueClassDegree(P): PlcFunElt → RngIntEltMinimum(P): PlcFunElt → RngEltResidueClassField(P): PlcFunElt → Rng, MapEvaluate(a, P): RngElt, PlcFunElt → RngEltLift(a, P): RngElt, PlcFunElt → FldFunEltLift(i, P): Infty, PlcFunElt → FldFunEltTwoGenerators(P): PlcFunElt → FldFunGElt, FldFunGEltLocalUniformizer(P): PlcFunElt → FldFunGEltUniformizingElement(P): PlcFunElt → FldFunGEltValuation(a, P): FldFunElt, PlcFunElt → RngIntEltIdeal(P): PlcFunElt → RngFunOrdIdlNorm(P): PlcFunElt → DivFunElt- Example: Counting Places of a Given Degree
- Completion at Places
- Divisors
- Creation of Structures
- Creation of Elements
- Related Structures
- Structure Invariants
- Structure Predicates
- Element Operations
- Arithmetic Operators
- D: DivFunElt → DivFunEltD1 + D2: DivFunElt, DivFunElt → DivFunEltD1 - D2: DivFunElt, DivFunElt → DivFunEltk * D: RngIntElt, DivFunElt → DivFunEltD div k: DivFunElt, RngIntElt → DivFunEltD mod k: DivFunElt, RngIntElt → DivFunEltP + D: PlcFunElt, DivFunElt → DivFunEltD + P: DivFunElt, PlcFunElt → DivFunEltD - P: DivFunElt, PlcFunElt → DivFunEltP - D: PlcFunElt, DivFunElt → DivFunEltQuotrem(D, k): DivFunElt, RngIntElt → DivFunElt, DivFunEltGCD(D1, D2): DivFunElt, DivFunElt → DivFunEltGcd(D1, D2): DivFunElt, DivFunElt → DivFunEltGreatestCommonDivisor(D1, D2): DivFunElt, DivFunElt → DivFunEltLCM(D1, D2): DivFunElt, DivFunElt → DivFunEltLcm(D1, D2): DivFunElt, DivFunElt → DivFunEltLeastCommonMultiple(D1, D2): DivFunElt, DivFunElt → DivFunElt
- Equality, Comparison, and Membership
- Predicates on Elements
- Other Element Operations
FunctionField(D): DivFunElt → FldFunDegree(D): DivFunElt → RngIntEltSupport(D): DivFunElt → [ PlcFunElt ]Numerator(D): DivFunElt → DivFunEltZeroDivisor(D): DivFunElt → DivFunEltDenominator(D): DivFunElt → DivFunEltPoleDivisor(D): DivFunElt → DivFunEltIdeals(D): DivFunElt → RngFunOrdIdl, RngFunOrdIdlNorm(D): DivFunElt → DivFunEltFiniteSplit(D): DivFunElt → DivFunElt, DivFunEltFiniteDivisor(D): DivFunElt → DivFunEltInfiniteDivisor(D): DivFunElt → DivFunEltDimension(D): DivFunElt → RngIntEltIndexOfSpeciality(D): DivFunElt → RngIntEltShortBasis(D : parameters): DivFunElt → [RngElt], [RngIntElt]Basis(D : parameters): DivFunElt → [ FldFunElt ]RiemannRochSpace(D): DivFunElt → ModFld, MapRiemannRochSpace(P): PlcFunElt → ModFld, MapValuation(D, P): DivFunElt, PlcFunElt → RngIntEltReduction(D): DivFunElt → DivFunElt, RngIntElt, DivFunElt, FldFunEltReduction(D, A): DivFunElt, DivFunElt → DivFunElt, RngIntElt, DivFunElt, FldFunEltGapNumbers(D, P): DivFunElt, PlcFunElt → SeqEnum[RngIntElt]GapNumbers(D): DivFunElt → SeqEnum[RngIntElt]- Example: Riemann-Roch Space of a Divisor
- Example: Algebraic Relation for Sine and Cosine
- Example: Sine and Cosine via Exponentials
RamificationDivisor(D): DivFunElt → DivFunEltWeierstrassPlaces(D): DivFunElt → [PlcFunElt]IsWeierstrassPlace(D, P): DivFunElt, PlcFunElt → BoolEltWronskianOrders(D): DivFunElt → [RngIntElt]ComplementaryDivisor(D): DivFunElt → DivFunEltDifferentialBasis(D): DivFunElt → [DiffFunElt]DifferentialSpace(D): DivFunElt → ModFld, MapParametrization(F, D): FldFun, DivFunElt → FldFunElt, [FldFunRatUElt]
- Functions related to Divisor Class Groups of Global Function Fields
ClassGroupGenerationBound(q, g): RngIntElt, RngIntElt → RngIntEltClassGroupGenerationBound(F): FldFunG → RngIntEltClassNumberApproximation(F, e): FldFunG, FldReElt → FldReEltClassNumberApproximationBound(q, g, e): RngIntElt, RngIntElt, FldReElt → RngIntEltClassGroup(F : parameters): FldFun → GrpAb, Map, MapClassGroupAbelianInvariants(F : parameters): FldFun → SeqEnumClassNumber(F): FldFun → RngIntElt- Example: Divisor Class Group Computations
GlobalUnitGroup(F): FldFun → GrpAb, MapIsGlobalUnit(a): FldFunElt → BoolEltIsGlobalUnitWithPreimage(a): FldFunElt → BoolElt, GrpAbEltPrincipalDivisorMap(F): FldFunG → MapClassGroupExactSequence(F): FldFunG → Map, Map, MapSUnitGroup(S): SetEnum[PlcFunElt] → GrpAb, MapIsSUnit(a, S): FldFunElt, SetEnum[PlcFunElt] → BoolEltIsSUnitWithPreimage(a, S): FldFunElt, SetEnum[PlcFunElt] → BoolElt, GrpAbEltSRegulator(S): SetEnum[PlcFunElt] → RngIntEltSPrincipalDivisorMap(S): SetEnum[PlcFunElt] → MapIsSPrincipal(D, S): DivFunElt, SetEnum[PlcFunElt] → BoolElt, FldFunEltSClassGroup(S): SetEnum[PlcFunElt] → GrpAb, Map, MapSClassGroupExactSequence(S): SetEnum[PlcFunElt] → Map, Map, MapSClassGroupAbelianInvariants(S): SetEnum[PlcFunElt] → SeqEnumSClassNumber(S): SetEnum[PlcFunElt] → RngIntEltClassGroupPRank(F): FldFunG → RngIntEltHasseWittInvariant(F): FldFunG → RngIntEltTateLichtenbaumPairing(D1, D2, m): DivFunElt, DivFunElt, RngIntElt → RngElt- Example: Tate-Lichtenbaum Pairing of Divisors
- 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 → RngIntEltDivisor(d): DiffFunElt → DivFunEltResidue(d, P): DiffFunElt, PlcFunElt → RngElt- Example: Functions on Differentials
Module(L, R): SeqEnum[ DiffFunElt ], Rng → Mod, Map, SeqEnum[ ModElt ]Relations(L, R): SeqEnum[ DiffFunElt ], Rng → ModTupRngRelations(L, R, m): SeqEnum[ DiffFunElt ], Rng, RngIntElt → ModTupRng- Example: Differentials as a Module
Cartier(b): DiffFunElt → DiffFunEltCartier(b, r): DiffFunElt, RngIntElt → DiffFunElt
- Other
- Weil Descent
WeilDescent(E,k): FldFun, FldFin → FldFunG, MapArtinSchreierExtension(c,a,b): FldFin, FldFin, FldFin → FldFunWeilDescentDegree(E,k): FldFun, FldFin → RngIntEltWeilDescentGenus(E,k): FldFun, FldFin → RngIntEltMultiplyFrobenius(b,f,F): RngElt, RngUPolElt, Map → RngElt- Example: GHS Weil Descent
- Function Field Database
- The Montes Algorithm
Montes(f, p): RngUPolElt, RngUPolElt → SeqEnum, SeqEnum, RngIntElt- Example: Montes Algorithm for a Polynomial
Montes(K, p): FldArith, RngElt- Example: Montes Algorithm for a Function Field
SFL(P, s): OMIdl, RngIntElt- Example: Refining an OM Prime Ideal Approximation
SetUseMontes(f): BoolEltSetUseMontes(t, f): Cat, BoolEltGetUseMontes(t): Cat → BoolEltSetVerbose("Montes", v): MonStgElt, RngIntElt
- Ideals in OM Representation
- Ideal Operations
pIntegralBasis(I, p): OMIdl, RngElt → SeqEnumSIntegralBasis(I, S): OMIdl, SeqEnum → SeqEnumSIntegralBasis(I, S): RngOrdFracIdl, [RngIntElt] → SeqEnumSIntegralBasis(I, S): RngFunOrdIdl, [RngUPolElt] → SeqEnumBasis(I): OMIdl → SeqEnum- Example: Integral Basis of an OM Ideal
TwoElement(I): OMIdl → FldArithElt, FldArithEltNorm(I): OMIdl → RngEltValuation(alpha, P : parameters): FldArithElt, OMIdl → RngIntElt, FldEltValuation(alpha, P : parameters): FldRatElt, OMIdl → RngIntElt, FldEltValuation(alpha, P : parameters): RngIntElt, OMIdl → RngIntElt, FldEltValuation(alpha, P : parameters): RngUPolElt, OMIdl → RngIntElt, FldEltValuation(I, P): OMIdl, OMIdl → RngIntElta mod P: FldArithElt, OMIdl → FldArithEltReduction(a, P): FldArithElt, OMIdl → FldArithEltReduction(a, P, m): FldArithElt, OMIdl, RngIntElt → [FldArithElt]Factorization(I): OMIdl → SeqEnumFactorisation(I): OMIdl → SeqEnum- Example: Creating an OM Ideal
ResidueField(I): OMIdl → FldDegree(I): OMIdl → RngIntElt- Example: Degree and Residue Field of OM Ideals
- Divisors in OM representation
- Bibliography