- Introduction
- Acknowledgement
- Creation Functions
- Creation of Algebraic Fields
- NumberField(f) : RngUPolElt -> FldNum
- RationalsAsNumberField() : -> FldNum
- NumberField(s) : [ RngUPolElt ] -> FldNum
- ext< F | s1, ..., sn > : FldAlg, RngUPolElt, ..., RngUPolElt -> FldAlg
- RadicalExtension(F, d, a) : Rng, RngIntElt, RngElt -> FldAlg
- SplittingField(F) : FldAlg -> FldAlg, SeqEnum
- SplittingField(f) : RngUPolElt -> FldAlg
- SplittingField(L) : [RngUPolElt] -> FldNum, [FldNumElt]
- sub< F | e1, ..., en > : FldAlg, FldAlgElt, ..., FldAlgElt -> FldAlg, Map
- MergeFields(F, L) : FldAlg, FldAlg -> SeqEnum
- Compositum(K, L) : FldAlg, FldAlg -> FldAlg
- Compositum(K, A) : FldAlg, FldAb -> FldAlg
- OptimizedRepresentation(F) : FldAlg -> FldAlg, Map
- Example RngOrd_opt-rep-ord (H39E1)
- Creation of Orders and Fields from Orders
- EquationOrder(f) : RngUPolElt -> RngOrd
- EquationOrder(S) : [RngUPolElt] -> RngOrd
- EquationOrder(K) : FldNum -> RngOrd
- SubOrder(O) : RngOrd -> RngOrd
- EquationOrder(O) : RngOrd -> RngOrd
- Integers(O) : RngOrd -> RngOrd
- Example RngOrd_Orders (H39E2)
- sub< O | a1, ..., ar > : RngOrd, RngOrdElt, ..., RngOrdElt -> RngOrd
- ext< O | a1, ..., ar > : RngOrd, RngOrdElt, ..., RngOrdElt -> RngOrd
- ext< Z | f > : RngInt, RngUPolElt -> RngOrd
- FieldOfFractions(O) : RngOrd -> FldOrd
- Order(F) : FldOrd -> RngOrd
- NumberField(O) : RngOrd -> FldNum
- NumberField(F) : FldOrd -> FldNum
- Example RngOrd_fractions (H39E3)
- OptimizedRepresentation(O) : RngOrd -> BoolElt, RngOrd, Map
- O + P : RngOrd, RngOrd -> RngOrd
- O meet P : RngOrd, RngOrd -> RngOrd
- AsExtensionOf(O, P) : RngOrd, RngOrd -> RngOrd
- Order(O, T, d) : RngOrd, AlgMatElt, RngIntElt -> RngOrd
- Order(O, M) : RngOrd, ModDed -> RngOrd
- Order( [ e1, ... en ] ): [FldAlgElt] -> RngOrd
- Maximal Orders
- Creation of Elements
- F ! a : FldAlg, RngElt -> FldAlgElt
- F ! [a0, a1, ..., am - 1] : FldAlg, [RngElt] -> FldAlgElt
- O ! a : RngOrd, RngElt -> RngOrdElt
- O ! [a0, a1, ..., am - 1] : RngOrd, [ RngElt ] -> RngOrdElt
- Random(F, m) : FldAlg, RngIntElt -> FldAlgElt
- Random(I, m) : RngOrdFracIdl, RngIntElt -> FldOrdElt
- Example RngOrd_Elements (H39E6)
- Creation of Homomorphisms
- hom< F -> R | r > : FldAlg, Rng, RngElt -> Map
- hom< O -> R | r > : RngOrd, Rng, RngElt -> Map
- Example RngOrd_Homomorphisms (H39E7)
- hom< O -> R | b1, ..., bn > : RngFunOrd, Rng, RngElt, ..., RngElt -> Map
- IsRingHomomorphism(m) : Map -> BoolElt
- Printing
- Real Precision
- Structure Operations
- General Functions
- Related Structures
- GroundField(F) : FldAlg -> Fld
- BaseRing(O) : RngOrd -> Rng
- AbsoluteField(F) : FldAlg -> FldAlg
- AbsoluteOrder(O) : RngOrd -> RngOrd
- SimpleExtension(F) : FldAlg -> FldAlg
- RelativeField(F, L) : FldAlg, FldAlg -> FldAlg
- Components(F) : FldAlg -> [FldAlg]
- Example RngOrd_Compositum (H39E8)
- Simplify(O) : RngOrd -> RngOrd
- LLL(O) : RngOrd -> RngOrd, AlgMatElt
- Example RngOrd_lll (H39E9)
- Embed(F, L, a) : FldAlg, FldAlg, FldAlgElt ->
- Embed(F, L, a) : FldAlg, FldAlg, [FldAlgElt] ->
- EmbeddingMap(F, L): FldAlg, FldAlg -> Map
- Example RngOrd_em (H39E10)
- Completion(K, P) : FldAlg, RngOrdIdl -> FldLoc, Map
- Completion(K, P) : FldAlg, PlcNumElt -> FldLoc, Map
- LocalRing(P, prec) : RngOrdIdl, RngIntElt -> RngLoc, Map
- Localization(O, P) : RngOrd, RngOrdIdl -> RngVal, Map
- Representing Fields as Vector Spaces
- Invariants
- Basis Representation
- Ring Predicates
- Order Predicates
- Field Predicates
- Setting Properties of Orders
- Element Operations
- Parent and Category
- Arithmetic
- w div v : RngOrdElt, RngOrdElt -> RngOrdElt
- Modexp(a, n, m) : RngOrdElt, RngIntElt, RngIntElt -> RngOrdElt
- Sqrt(a) : RngOrdElt -> RngOrdElt
- Root(a, n) : RngOrdElt, RngIntElt -> RngOrdElt
- IsPower(a, k) : FldAlgElt, RngIntElt -> BoolElt, FldAlgElt
- Denominator(a) : FldAlgElt -> RngIntElt
- Numerator(a) : FldAlgElt -> RngIntElt
- Qround(E, M): FldAlgElt, RngIntElt -> FldAlgElt
- Equality and Membership
- Predicates on Elements
- Field Generators
- Real and Complex Embeddings
- Heights
- Norm, Trace, and Minimal Polynomial
- The Quadratic Defect
- Other Functions
- ElementToSequence(a) : FldAlgElt -> [ FldAlgElt ]
- Eltseq(E, k) : FldAlgElt, FldAlg -> [RngElt]
- Flat(e) : FldAlgElt -> [FldRatElt]
- a[i] : FldAlgElt, RngIntElt -> FldRatElt
- ProductRepresentation(a) : RngOrdElt -> [ RngOrdElt ], [ RngIntElt ]
- ProductRepresentation(P, E) : [ FldAlgElt ], [ RngIntElt ] -> FldAlgElt
- Valuation(w, I) : RngOrdElt, RngOrdIdl -> RngIntElt
- Decomposition(a) : RngOrdElt -> SeqEnum[<RngOrdIdl, RngIntElt>]
- Divisors(a) : RngOrdElt -> SeqEnum[RngOrdElt]
- Index(a) : RngOrdElt -> RngIntElt
- Different(a) : RngOrdElt -> RngOrdElt
- DegreeOnePrimeIdeals(O, B) : RngOrd, RngIntElt -> [ RngOrdIdl ]
- Ideal Class Groups
- Unit Groups
- UnitRank(O) : RngOrd -> RngIntElt
- TorsionUnitGroup(O) : RngOrd -> GrpAb, Map
- UnitGroup(O) : RngOrd -> GrpAb, Map
- UnitGroup(K) : FldNum -> GrpAb, Map
- IndependentUnits(O) : RngOrd -> GrpAb, Map
- pFundamentalUnits(O, p) : RngOrd, RngIntElt -> GrpAb, Map
- UnitGroupAsSubgroup(O) : RngOrd -> GrpAb
- MergeUnits(K, a) : FldNum, FldNumElt -> BoolElt
- Example RngOrd_UnitGroup (H39E20)
- IsExceptionalUnit(u) : RngOrdElt -> BoolElt
- ExceptionalUnitOrbit(u) : RngOrdElt -> [ RngOrdElt ]
- ExceptionalUnits(O) : RngOrd -> [ RngOrdElt ]
- UnitsWithSigns(O, oo, Signs) : RngOrd, [ PlcNumElt ], [ RngInt ] -> [ RngOrdElt ]
- UnitsWithSigns(O, x) : RngOrd, RngElt -> [ RngOrdElt ]
- UnitsWithSigns(x) : RngOrdElt -> [ RngOrdElt ]
- HasTotallyPositiveGenerator(I) : RngOrdFracIdl -> BoolElt, [ RngOrdElt ]
- Diophantine Equations
- Norm Equations
- NormEquation(O, m) : RngOrd, RngIntElt -> BoolElt, [ RngOrdElt ]
- NormEquation(F, m) : FldAlg, RngIntElt -> BoolElt, [ FldAlgElt ]
- NormEquation(m, N): RngElt, Map -> BoolElt, RngElt
- IntegralNormEquation(a, N, O) : RngElt, Map, RngOrd -> BoolElt, [RngOrdElt]
- SimNEQ(K, e, f) : FldNum, FldNumElt, FldNumElt -> BoolElt, [FldNumElt]
- Example RngOrd_norm-equation (H39E21)
- Thue Equations
- Unit Equations
- Index Form Equations
- Ideals and Quotients
- Creation of Ideals in Orders
- Invariants
- Basis Representation
- Two--Element Presentations
- Standard Names
- Predicates on Ideals
- IsIntegral(I) : RngOrdFracIdl -> BoolElt
- IsZero(I) : RngOrdFracIdl -> BoolElt
- IsOne(I) : RngOrdIdl -> BoolElt
- IsPrime(I) : RngOrdIdl -> BoolElt, RngOrdIdl
- IsPrincipal(I) : RngOrdFracIdl -> BoolElt, FldOrdElt
- IsRamified(P) : RngOrdIdl -> BoolElt
- IsRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
- IsTotallyRamified(P) : RngOrdIdl -> BoolElt
- IsTotallyRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
- IsTotallyRamified(K) : FldAlg -> BoolElt
- IsTotallyRamified(O) : RngOrd -> BoolElt
- IsWildlyRamified(P) : RngOrdIdl -> BoolElt
- IsWildlyRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
- IsTamelyRamified(P) : RngOrdIdl -> BoolElt
- IsTamelyRamified(P, O) : RngOrdIdl, RngOrd -> BoolElt
- IsUnramified(P) : RngOrdIdl -> BoolElt
- IsUnramified(P, O) : RngOrdIdl, RngOrd -> BoolElt
- IsInert(P) : RngOrdIdl -> BoolElt
- IsInert(P, O) : RngOrdIdl, RngOrd -> BoolElt
- IsSplit(P) : RngOrdIdl -> BoolElt
- IsSplit(P, O) : RngOrdIdl, RngOrd -> BoolElt
- IsTotallySplit(P) : RngOrdIdl -> BoolElt
- IsTotallySplit(P, O) : RngOrdIdl, RngOrd -> BoolElt
- Ideal Arithmetic
- I * J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
- x * I : RngElt, RngOrdFracIdl -> RngOrdFracIdl
- &* L : [RngOrdFracIdl] -> RngOrdFracIdl
- I div J : RngOrdIdl, RngOrdIdl -> RngOrdIdl
- I / J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
- I / x : RngOrdFracIdl, RngElt -> RngOrdFracIdl
- I + J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
- I ^ k : RngOrdFracIdl, RngIntElt -> RngOrdFracIdl
- I eq J : RngOrdFracIdl, RngOrdFracIdl -> BoolElt
- I subset J : RngOrdIdl, RngOrdIdl -> BoolElt
- E in I: RngOrdElt, RngOrdIdl -> BoolElt
- LCM(I, J) : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
- GCD(I, J) : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
- Content(M) : Mtrx -> RngOrdFracIdl
- I meet J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
- &meet S : [RngOrdFracIdl] -> RngOrdFracIdl
- I meet R : RngOrdFracIdl, Rng -> Any
- a mod I : RngOrdElt, RngOrdIdl -> RngOrdElt
- InverseMod(E, M) : RngOrdElt, RngIntElt -> RngOrdElt
- ColonIdeal(I, J) : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
- Example RngOrd_colon (H39E29)
- IntegralSplit(I) : RngOrdFracIdl -> RngOrdIdl, RngElt
- Different(I) : RngOrdFracIdl -> RngOrdFracIdl
- Codifferent(I) : RngOrdFracIdl -> RngOrdFracIdl
- Roots of Ideals
- Factorization and Primes
- Decomposition(O, p) : RngOrd, RngIntElt -> [<RngOrdIdl, RngIntElt>]
- DecompositionType(O, p) : RngOrd, RngIntElt -> [<RngIntElt, RngIntElt>]
- Factorization(I) : RngOrdFracIdl -> [<RngOrdIdl, RngIntElt>]
- Example RngOrd_non-maximal-fact (H39E30)
- Divisors(I) : RngOrdIdl -> [<RngOrdIdl, RngIntElt>]
- Support(I) : RngOrdFracIdl -> RngOrdIdl
- Support(L) : [RngOrdFracIdl] -> RngOrdIdl
- CoprimeBasis(L) : [RngOrdFracIdl] -> RngOrdIdl
- CoprimeBasisInsert(~L, I) : [RngOrdIdl], RngOrdFracIdl ->
- PowerProduct(B, E) : [RngOrdFracIdl], [RngIntElt] -> RngOrdFracIdl
- Other Ideal Operations
- ChineseRemainderTheorem(I1, I2, e1, e2) : RngOrdIdl, RngOrdIdl, RngOrdElt, RngOrdElt -> RngOrdElt
- CRT(I1, L1, e1, L2) : RngOrdIdl, [RngIntElt], RngOrdElt, [RngIntElt] -> RngOrdElt
- WeakApproximation(I, V) : [RngOrdIdl], [RngIntElt] -> FldOrdElt
- Idempotents(I, J) : RngOrdIdl, RngOrdIdl -> BoolElt, RngOrdElt, RngOrdElt
- CoprimeRepresentative(I, J) : RngOrdIdl, RngOrdIdl -> FldOrdElt
- IdealsUpTo(B, O) : RngIntElt, RngOrd -> [RngOrdIdl]
- ClassRepresentative(I) : RngOrdFracIdl -> RngOrdFracIdl
- SUnitGroup(I) : RngOrdFracIdl -> GrpAb, Map
- Example RngOrd_S-Units (H39E31)
- SUnitAction(SU, Act, S) : Map, Map, SeqEnum[RngOrdIdl] -> Map
- SUnitAction(SU, Act, S) : Map, SeqEnum[Map], SeqEnum[RngOrdIdl] -> [Map]
- SUnitDiscLog(SU, x, S) : Map, FldAlgElt, SeqEnum[RngOrdIdl] -> GrpAbElt
- Example RngOrd_S-Units, advanced (H39E32)
- Quotient Rings
- Places and Divisors
- The Montes Algorithm
- Montes(f, p) : RngUPolElt, RngElt -> SeqEnum, SeqEnum, RngIntElt
- Example RngOrd_montes-eg-1 (H39E35)
- Montes(K, p) : FldArith, RngElt ->
- Example RngOrd_montes-eg-2 (H39E36)
- SFL(P, s) : OMIdl, RngIntElt ->
- Example RngOrd_sfl (H39E37)
- 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 RngOrd_om-ideal-op (H39E40)
- 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 RngOrd_om-ideal-ops (H39E41)
- ResidueField(I) : OMIdl -> Fld
- Example RngOrd_om-ideals-deg-res (H39E42)
- Bibliography
V2.28, 13 July 2023