NUMBER FIELDS AND ORDERS
Acknowledgements
Introduction
Types
Acknowledgement
Creation Functions
Creation of Algebraic Fields
Creation of Orders and Fields from Orders
Maximal Orders
Orders and Ideals
Creation of Elements
Creation of Homomorphisms
Printing
Real Precision
Structure Operations
General Functions
Related Structures
Representing Fields as Vector Spaces
Invariants
Basis Representation
Ring Predicates
Order Predicates
Field Predicates
Setting Properties of Orders
Element Operations
Parent and Category
Arithmetic
Equality and Membership
Predicates on Elements
Field Generators
Real and Complex Embeddings
Heights
Norm, Trace, and Minimal Polynomial
The Quadratic Defect
Other Functions
Ideal Class Groups
Class Group Internals
Setting the Class Group Bounds
Class Group Map Caching
Unit Groups
Diophantine Equations
Norm Equations
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
Ideal Arithmetic
Roots of Ideals
Factorization and Primes
Other Ideal Operations
Quotient Rings
Operations on Quotient Rings
Elements of Quotients
Reconstruction
Places and Divisors
Creation of Structures
Operations on Structures
Creation of Elements
Arithmetic with Places and Divisors
Other Functions for Places and Divisors
The Montes Algorithm
Ideals in OM Representation
Ideal Arithmetic
Ideal Predicates
Ideal Operations
Bibliography
Introduction
Types
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
MaximalOrder(O) : RngOrd -> RngOrd
MaximalOrder(f) : RngUPolElt -> RngOrd
Example RngOrd_max_order (H39E4)
Orders and Ideals
pMaximalOrder(O, p) : RngOrd, RngIntElt -> RngOrd
pRadical(O, p) : RngOrd, RngIntElt -> RngOrdIdl
MultiplicatorRing(I) : RngOrdFracIdl -> Rng
Example RngOrd_Round2 (H39E5)
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
SetVerbose(s, n) : MonStgElt, RngIntElt ->
SetKantPrinting(f) : BoolElt -> BoolElt
Real Precision
SetKantPrecision(F, n) : FldAlg, RngIntElt ->
Structure Operations
General Functions
AssignNames(~K, s) : FldNum, [ MonStgElt ] ->
Name(K, i) : FldNum, RngIntElt -> FldNumElt
AssignNames(~F, s) : FldOrd, [ MonStgElt ] ->
F . i : FldOrd, RngIntElt -> FldOrdElt
O . i : RngOrd, RngIntElt -> FldOrdElt
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
HasEmbedding(F, L) : FldAlg, FldAlg -> Bool
CanComputeEmbedding(F, L) : FldAlg, FldAlg -> Bool
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
Algebra(K, J) : FldAlg, Fld -> AlgAss, Map
VectorSpace(K, J) : FldAlg, Fld -> ModTupFld, Map
Example RngOrd_vector_space_eg (H39E11)
Invariants
Degree(O) : RngOrd -> RngIntElt
AbsoluteDegree(O) : RngOrd -> RngIntElt
Discriminant(O) : RngOrd -> RngIntElt
AbsoluteDiscriminant(O) : RngOrd -> RngIntElt
AbsoluteDiscriminant(K) : FldAlg -> FldRatElt
ReducedDiscriminant(O) : RngOrd -> RngIntElt
Regulator(O: parameters) : RngOrd -> FldReElt
RegulatorLowerBound(O) : RngOrd -> FldReElt
Signature(O) : RngOrd -> RngIntElt, RngIntElt
UnitRank(O) : RngOrd -> RngIntElt
Index(O, S) : RngOrd, RngOrd -> RngIntElt
DefiningPolynomial(F) : FldAlg -> RngUPolElt
Zeroes(O, n) : RngOrd, RngIntElt -> [ FldReElt ]
Example RngOrd_zero (H39E12)
Different(O) : RngOrd -> RngOrdIdl
Conductor(O) : RngOrd -> RngOrdIdl
Basis Representation
Basis(O) : RngOrd -> [ FldOrdElt ]
IntegralBasis(F) : FldAlg -> [ FldAlgElt ]
Example RngOrd_basis-ring (H39E13)
AbsoluteBasis(K) : FldAlg -> [FldAlgElt]
BasisMatrix(O) : RngOrd -> AlgMatElt
TransformationMatrix(O, P) : RngOrd, RngOrd -> AlgMatElt, RngIntElt
CoefficientIdeals(O) : RngOrd -> [RngOrdFracIdl]
Example RngOrd_Bases (H39E14)
MultiplicationTable(O) : RngOrd -> [AlgMatElt]
TraceMatrix(O) : RngOrd -> AlgMatElt
Example RngOrd_MultiplicationTable (H39E15)
Ring Predicates
N eq O : RngOrd, RngOrd -> BoolElt
F eq L : FldAlg, FldAlg -> BoolElt
IsNumberField(R) : . -> BoolElt
IsAlgebraicField(R) : Any -> BoolElt
IsEuclideanDomain(F) : FldAlg -> BoolElt
IsSimple(F) : FldAlg -> BoolElt
IsPrincipalIdealRing(F) : FldAlg -> BoolElt
IsPrincipalIdealRing(O) : RngOrd -> BoolElt
HasComplexConjugate(K) : FldAlg -> BoolElt, Map
ComplexConjugate(x) : FldAlgElt -> FldAlgElt
Order Predicates
IsEquationOrder(O) : RngOrd -> BoolElt
IsMaximal(O) : RngOrd -> BoolElt
IsAbsoluteOrder(O) : RngOrd -> BoolElt
IsWildlyRamified(O) : RngOrd -> BoolElt
IsTamelyRamified(O) : RngOrd -> BoolElt
IsUnramified(O) : RngOrd -> BoolElt
Field Predicates
IsIsomorphic(F, L) : FldAlg, FldAlg -> BoolElt, Map
IsSubfield(F, L) : FldAlg, FldAlg -> BoolElt, Map
IsNormal(F) : FldAlg -> BoolElt
IsAbelian(F) : FldAlg -> BoolElt
IsCyclic(F) : FldAlg -> BoolElt
IsAbsoluteField(K) : FldAlg -> BoolElt
IsWildlyRamified(K) : FldAlg -> BoolElt
IsTamelyRamified(K) : FldAlg -> BoolElt
IsUnramified(K) : FldAlg -> BoolElt
IsQuadratic(K) : FldAlg -> BoolElt, FldQuad
IsTotallyReal(K) : FldAlg -> BoolElt
Setting Properties of Orders
SetOrderMaximal(O, b) : RngOrd, BoolElt ->
SetOrderTorsionUnit(O, e, r) : RngOrd, RngOrdElt, RngIntElt ->
SetOrderUnitsAreFundamental(O) : RngOrd ->
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
IsIntegral(a) : FldAlgElt -> BoolElt
IsPrimitive(a) : FldAlgElt -> BoolElt
IsTorsionUnit(w) : RngOrdElt -> BoolElt
IsPower(w, n) : RngOrdElt, RngIntElt -> BoolElt, RngOrdElt
IsTotallyPositive(a) : RngOrdElt -> BoolElt
Field Generators
K . 1 : FldNum -> FldNumElt
PrimitiveElement(K) : FldNum -> FldNumElt
PrimitiveElement(O) : RngOrd -> RngOrdElt
Generators(K): FldAlg -> [FldAlgElt]
Generators(K, k) : FldAlg, FldAlg -> [FldAlgElt]
Real and Complex Embeddings
Conjugates(a) : FldAlgElt -> [ FldComElt ]
Conjugate(a, l) : FldAlgElt, RngIntElt -> FldReElt
Conjugate(a, l) : FldAlgElt, [RngIntElt] -> FldReElt
AbsoluteValues(a) : FldAlgElt -> [FldReElt]
Logs(a) : FldAlgElt -> [FldReElt]
InfinitePlaces(K) : FldAlg -> [PlcNumElt]
Evaluate(x, p) : FldAlgElt, PlcNumElt -> RngElt
RealEmbeddings(a) : FldAlgElt -> []
MinkowskiLattice(O) : RngOrd -> Lat, Map
MinkowskiLattice(I) : RngOrdIdl -> Lat, Map
MinkowskiSpace(F) : FldAlg -> Lat, Map
Heights
AbsoluteLogarithmicHeight(a) : FldAlgElt -> FldReElt
CoefficientHeight(E) : RngOrdElt -> RngIntElt
CoefficientLength(E) : RngOrdElt -> RngIntElt
Length(a) : FldAlgElt -> FldReElt
Example RngOrd_Discriminant (H39E16)
Norm, Trace, and Minimal Polynomial
Norm(a) : FldAlgElt -> FldAlgElt
AbsoluteNorm(a) : FldAlgElt -> FldRatElt
Trace(a) : FldAlgElt -> FldAlgElt
AbsoluteTrace(a) : FldAlgElt -> FldRatElt
CharacteristicPolynomial(a) : FldAlgElt -> RngUPolElt
AbsoluteCharacteristicPolynomial(a) : FldAlgElt -> RngUPolElt
MinimalPolynomial(a) : FldAlgElt -> RngUPolElt
AbsoluteMinimalPolynomial(a) : FldAlgElt -> RngUPolElt
RepresentationMatrix(a) : FldAlgElt -> AlgMatElt
AbsoluteRepresentationMatrix(a) : FldAlgElt -> AlgMatElt
Example RngOrd_NormsEtc (H39E17)
The Quadratic Defect
QuadraticDefect(a, p) : RngElt, RngOrdIdl -> RngIntElt
IsLocalSquare(a, p) : RngElt, RngOrdIdl -> BoolElt
LocalMultiplicativeGroupModSquares(p) : RngOrdIdl -> ModFld, Map
UnitSquareClassReps(p) : RngOrdIdl -> SeqEnum
NiceUnitSquareClassRepresentative(u, p) : RngElt, RngOrdIdl -> RngElt
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
SetPrintClassGroupWarnings(b) : BoolElt ->
ClassGroup(O: parameters) : RngOrd -> GrpAb, Map
RingClassGroup(O) : RngOrd -> GrpAb, Map
ConditionalClassGroup(O) : RngOrd -> GrpAb, Map
ClassGroupPrimeRepresentatives(O, I) : RngOrd, RngOrdIdl -> Map
ClassNumber(O: parameters) : RngOrd -> RngIntElt
MinkowskiBound(K) : FldNum -> RngIntElt
BachBound(K) : FldNum -> RngIntElt
GRHBound(K) : FldNum -> RngIntElt
FactorBasisVerify(O, a, b) : RngOrd, RngIntElt, RngIntElt ->
Class Group Internals
EulerProduct(O, B) : RngOrd, RngIntElt -> FldReElt
ResidueGRH(O, B) : RngOrd, RngIntElt -> FldReElt
ResidueGRHbound(O, e) : RngOrd, FldReElt -> FldReElt
FactorBasis(K, B) : FldNum, RngIntElt -> [ RngOrdIdl ]
FactorBasis(O) : RngOrd -> [ RngOrdIdl ], Integer
RelationMatrix(O) : RngOrd -> ModHomElt
Relations(O) : RngOrd -> ModHomElt
ClassGroupCyclicFactorGenerators(O) : RngOrd -> ModHomElt
Example RngOrd_ClassGroup (H39E18)
Setting the Class Group Bounds
SetClassGroupBounds(string) : MonStgElt ->
SetClassGroupBounds(n) : RngIntElt ->
Example RngOrd_class-group-bounds (H39E19)
Class Group Map Caching
ClassGroupGetUseMemory(O) : RngOrd -> BoolElt
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
Thue(f) : RngUPolElt -> Thue
Thue(O) : RngOrd -> Thue
Evaluate(t, a, b) : Thue, RngIntElt, RngIntElt -> RngIntElt
Solutions(t, a) : Thue, RngIntElt -> [ [ RngIntElt, RngIntElt ] ]
Example RngOrd_thue (H39E22)
Unit Equations
UnitEquation(a, b, c) : FldNumElt, FldNumElt, FldNumElt -> [ ModHomElt ]
Example RngOrd_uniteq (H39E23)
Index Form Equations
IndexFormEquation(O, k) : RngOrd, RngIntElt -> [ RngOrdElt ]
Example RngOrd_index-form (H39E24)
Ideals and Quotients
Creation of Ideals in Orders
x * O : RngElt, RngOrd -> RngOrdFracIdl
F !! I : RngOrd, RngInt -> RngOrdFracIdl
ideal< O | a1, a2, ... , am > : RngOrd, RngElt, ..., RngElt -> RngOrdFracIdl
Example RngOrd_Ideals (H39E25)
FractionalIdeal(x): FldRatElt -> RngIntFracIdl
FractionalIdeal(I) : RngInt -> RngIntFracIdl
Invariants
Order(I) : RngOrdFracIdl -> RngOrd
Denominator(I) : RngOrdFracIdl -> RngIntElt
PrimitiveElement(I) : RngOrdIdl -> RngOrdElt
Index(O, I) : RngOrd, RngOrdIdl -> RngIntElt
Norm(I) : RngOrdIdl -> RngIntElt
MinimalInteger(I) : RngOrdIdl -> RngElt
Minimum(I) : RngOrdFracIdl -> RngElt
AbsoluteNorm(I) : RngOrdIdl -> RngIntElt
CoefficientHeight(I) : RngOrdIdl -> RngIntElt
CoefficientLength(I) : RngOrdIdl -> RngIntElt
RamificationIndex(I, p) : RngOrdIdl, RngIntElt -> RngIntElt
RamificationDegree(I) : RngOrdIdl -> RngIntElt
AbsoluteRamificationDegree(I) : RngOrdIdl -> RngIntElt
ResidueClassField(O, I) : RngOrd, RngOrdIdl -> FldFin, Map
Degree(I) : RngOrdIdl -> RngIntElt
AbsoluteInertiaDegree(I) : RngOrdIdl -> RngIntElt
Valuation(I, p) : RngOrdFracIdl, RngOrdIdl -> RngIntElt
Content(I) : RngOrdFracIdl -> RngIntElt
Example RngOrd_ideal-invar (H39E26)
Basis Representation
Basis(I) : RngOrdIdl -> [RngOrdElt]
BasisMatrix(I) : RngOrdFracIdl -> MtrxSpcElt
TransformationMatrix(I) : RngOrdFracIdl -> MtrxSpcElt, RngIntElt
CoefficientIdeals(I) : RngOrdFracIdl -> [RngOrdFracIdl]
Example RngOrd_ideal-basis (H39E27)
Module(I) : RngOrdFracIdl -> ModDed, Map
Two--Element Presentations
Generators(I) : RngOrdIdl -> [ RngOrdElt ]
TwoElement(I) : RngOrdFracIdl -> FldOrdElt, FldOrdElt
TwoElementNormal(I) : RngOrdIdl -> RngOrdElt, RngOrdElt, RngIntElt
Example RngOrd_ideal-two (H39E28)
Standard Names
LMFDBLabel(I) : RngOrdIdl -> MonStgElg
LMFDBIdeal(K, s) : FldNum, MonStgElt -> RngOrdIdl
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
IsSquarefree(I) : RngOrdIdl -> 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
Root(I, k) : RngOrdFracIdl, RngIntElt -> RngOrdFracIdl
IsPower(I, k) : RngOrdFracIdl, RngIntElt -> BoolElt, RngOrdFracIdl
SquareRoot(I) : RngOrdFracIdl -> RngOrdFracIdl
IsSquare(I) : RngOrdFracIdl -> BoolElt, RngOrdFracIdl
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
Operations on Quotient Rings
quo< O | I > : RngOrd, RngOrdIdl -> RngOrdRes
quo< O | m > : RngOrd, RngIntElt -> RngOrdRes
UnitGroup(OQ) : RngOrdRes -> GrpAb, Map
Modulus(OQ) : RngOrdRes -> RngOrdIdl
Example RngOrd_quotient (H39E33)
Elements of Quotients
OQ ! a : RngOrdRes, Elt -> RngOrdResElt
Random(OQ) : RngOrdRes -> RngOrdResElt
a mod I : RngOrdElt, RngOrdIdl -> RngOrdElt
IsZero(a) : RngOrdResElt -> BoolElt
IsOne(a) : RngOrdResElt -> BoolElt
IsMinusOne(a) : RngOrdResElt -> BoolElt
IsUnit(a) : RngOrdResElt -> BoolElt
Eltseq(a) : RngOrdResElt -> []
EuclideanNorm(a) : RngOrdResElt -> RngIntElt
Annihilator(a) : RngOrdResElt -> RngOrdResElt
Reconstruction
ReconstructionEnvironment(p, k) : RngOrdIdl, RngIntElt -> RngOrdRecoEnv
Reconstruct(x, R) : RngOrdElt, RngOrdRecoEnv -> RngOrdElt
ChangePrecision(~R, k) : RngOrdRecoEnv, RngIntElt ->
Example RngOrd_order-reco (H39E34)
Places and Divisors
Creation of Structures
Places(K) : FldNum -> PlcNum
Operations on Structures
NumberField(P) : PlcNum -> FldNum
Creation of Elements
Place(I) : RngOrdIdl -> PlcNumElt
Decomposition(K, I) : FldAlg, Infty -> SeqEnum
Decomposition(K, p) : FldNum, PlcNumElt -> SeqEnum
Decomposition(m, p) : Map[FldRat, FldAlg], RngIntElt -> SeqEnum[<PlcNumElt, RngIntElt>]
InfinitePlaces(K) : FldAlg -> [PlcNumElt]
RealPlaces(K) : FldAlg -> [PlcNumElt]
Divisor(pl) : PlcNumElt -> DivNumElt
Divisor(I) : RngOrdFracIdl -> DivNumElt
Divisor(x) : FldNumElt -> DivNumElt
Arithmetic with Places and Divisors
Other Functions for Places and Divisors
Valuation(a, p) : FldNumElt, PlcNumElt -> RngElt
Valuation(I, p) : RngOrdFracIdl , PlcNumElt -> RngElt
Support(D) : DivNumElt -> SeqEnum, SeqEnum
Ideal(D) : DivNumElt -> RngOrdIdl
IsFinite(p) : PlcNumElt -> BoolElt
IsInfinite(p) : PlcNumElt -> BoolElt, RngIntElt
IsReal(p) : PlcNumElt -> BoolElt
IsComplex(p) : PlcNumElt -> BoolElt
Extends(P, p) : PlcNumElt, PlcNumElt -> BoolElt
InertiaDegree(P) : PlcNumElt -> RngIntElt
Degree(D) : DivNumElt -> RngElt
NumberField(P) : PlcNumElt -> FldNum
ResidueClassField(P) : PlcNumElt -> Fld
UniformizingElement(P) : PlcNumElt -> FldNumElt
LocalDegree(P) : PlcNumElt -> RngIntElt
RamificationIndex(P) : PlcNumElt -> RngIntElt
DecompositionGroup(P) : PlcNumElt -> GrpPerm
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(I) : OMIdl -> RngOrdIdl
OMRepresentation(I) : RngFunOrdIdl -> OMIdl
OMRepresentation(L, S) : FldArith, [FldArithElt] -> OMIdl
Example RngOrd_om-rep-ideal (H39E38)
Ideal Arithmetic
I + J : OMIdl, OMIdl -> OMIdl
I ^ n : OMIdl, RngIntElt -> OMIdl
Example RngOrd_om-ideal-arith (H39E39)
Ideal Predicates
IsOne(I) : OMIdl -> BoolElt
IsZero(I) : OMIdl -> BoolElt
I eq J : OMIdl, OMIdl -> BoolElt
a in I : RngElt, OMIdl -> BoolElt
I subset J : OMIdl, OMIdl -> BoolElt
IsPrime(I) : OMIdl -> BoolElt
IsIntegral(I) : OMIdl -> BoolElt
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
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