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Acknowledgements
Introduction
Creation Functions
Creation of General Algebraic Fields
Creation of Orders and Fields from Orders
Maximal Orders
Orders and Ideals
Creation of Elements
Creation of Homomorphisms
Special Options
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
Finding Special Elements
Real and Complex Valued Functions
Norm, Trace, and Minimal Polynomial
Other Functions
Ideal Class Groups
Setting the Class Group Bounds Globally
Unit Groups
Solving Equations
Norm Equations
Thue Equations
Unit Equations
Index Form Equations
Ideals and Quotients
Creation of Ideals in Orders
Invariants
Basis Representation
Two--Element Presentations
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
Bibliography
DETAILS
Introduction
Creation Functions
Creation of General Algebraic Fields
NumberField(f) : RngUPolElt -> FldNum
NumberField(Q) : FldRat -> 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 (H34E1)
Creation of Orders and Fields from Orders
EquationOrder(f) : RngUPolElt -> RngOrd
EquationOrder(K) : FldNum -> RngOrd
SubOrder(O) : RngOrd -> RngOrd
EquationOrder(O) : RngOrd -> RngOrd
Integers(O) : RngOrd -> RngOrd
Example RngOrd_Orders (H34E2)
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 (H34E3)
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 (H34E4)
Orders and Ideals
pMaximalOrder(O, p) : RngOrd, RngIntElt -> RngOrd
pRadical(O, p) : RngOrd, RngIntElt -> RngOrdIdl
MultiplicatorRing(I) : RngOrdFracIdl -> Rng
Example RngOrd_Round2 (H34E5)
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 (H34E6)
Creation of Homomorphisms
hom< F -> R | r > : FldAlg, Rng, RngElt -> Map
hom< O -> R | r > : RngOrd, Rng, RngElt -> Map
Example RngOrd_Homomorphisms (H34E7)
hom< O -> R | b1, ..., bn > : RngFunOrd, Rng, RngElt, ..., RngElt -> Map
IsRingHomomorphism(m) : Map -> BoolElt
Special Options
SetVerbose(s, n) : MonStgElt, RngIntElt ->
SetKantPrinting(f) : BoolElt -> BoolElt
SetKantPrecision(n) : 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 -> RngOrdElt
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
Example RngOrd_Compositum (H34E8)
Simplify(O) : RngOrd -> RngOrd
LLL(O) : RngOrd -> RngOrd, AlgMatElt
Example RngOrd_lll (H34E9)
Embed(F, L, a) : FldAlg, FldAlg, FldAlgElt ->
Embed(F, L, a) : FldAlg, FldAlg, [FldAlgElt] ->
EmbeddingMap(F, L): FldAlg, FldAlg -> Map
Example RngOrd_em (H34E10)
Lattice(O) : RngOrd -> Lat, Map
MinkowskiSpace(F) : FldAlg -> Lat, Map
Completion(K, P) : FldAlg, RngOrdIdl -> FldLoc, Map
Completion(K, P) : FldAlg, PlcNumElt -> FldLoc, Map
LocalRing(P, prec) : RngOrdIdl, RngIntElt -> RngLoc, 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 (H34E11)
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 -> FldPrElt
RegulatorLowerBound(O) : RngOrd -> FldPrElt
Signature(O) : RngOrd -> RngIntElt, RngIntElt
UnitRank(O) : RngOrd -> RngIntElt
Index(O, S) : RngOrd, RngOrd -> RngIntElt
DefiningPolynomial(F) : FldAlg -> RngUPolElt
Zeroes(O, n) : RngOrd, RngIntElt -> [ FldPrElt ]
Example RngOrd_zero (H34E12)
Different(O) : RngOrd -> RngOrdIdl
Conductor(O) : RngOrd -> RngOrdIdl
Basis Representation
Basis(O) : RngOrd -> [ FldOrdElt ]
IntegralBasis(F) : FldAlg -> [ FldAlgElt ]
Example RngOrd_basis-ring (H34E13)
AbsoluteBasis(K) : FldAlg -> [FldAlgElt]
BasisMatrix(O) : RngOrd -> AlgMatElt
TransformationMatrix(O, P) : RngOrd, RngOrd -> AlgMatElt, RngIntElt
CoefficientIdeals(O) : RngOrd -> [RngOrdFracIdl]
Example RngOrd_Bases (H34E14)
MultiplicationTable(O) : RngOrd -> [AlgMatElt]
TraceMatrix(O) : RngOrd -> AlgMatElt
Example RngOrd_MultiplicationTable (H34E15)
Ring Predicates
N eq O : RngOrd, RngOrd -> BoolElt
F eq L : FldAlg, FldAlg -> 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
Finding Special Elements
K . 1 : FldNum -> FldNumElt
PrimitiveElement(K) : FldNum -> FldNumElt
Generators(K): FldAlg -> [FldAlgElt]
Generators(K, k) : FldAlg, FldAlg -> [FldAlgElt]
PrimitiveElement(O) : RngOrd -> RngOrdElt
Real and Complex Valued Functions
AbsoluteValues(a) : FldAlgElt -> [FldPrElt]
AbsoluteLogarithmicHeight(a) : FldAlgElt -> FldPrElt
Conjugates(a) : FldAlgElt -> [ FldComElt ]
Conjugate(a, k) : FldAlgElt, RngIntElt -> FldPrElt
Conjugate(a, l) : FldAlgElt, [RngIntElt] -> FldReElt
Length(a) : FldAlgElt -> FldPrElt
Logs(a) : FldAlgElt -> [FldPrElt]
CoefficientHeight(E) : RngOrdElt -> RngIntElt
CoefficientLength(E) : RngOrdElt -> RngIntElt
Example RngOrd_Discriminant (H34E16)
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 (H34E17)
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
Ideal Class Groups
DegreeOnePrimeIdeals(O, B) : RngOrd, RngIntElt -> [ RngOrdIdl ]
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
BachBound(K) : FldNum -> RngIntElt
MinkowskiBound(K) : FldNum -> RngIntElt
FactorBasis(K, B) : FldNum, RngIntElt -> [ RngOrdIdl ]
FactorBasis(O) : RngOrd -> [ RngOrdIdl ], Integer
RelationMatrix(K, B) : FldNum, RngIntElt -> ModHomElt
RelationMatrix(O) : RngOrd -> ModHomElt
Relations(O) : RngOrd -> ModHomElt
ClassGroupCyclicFactorGenerators(O) : RngOrd -> ModHomElt
Example RngOrd_ClassGroup (H34E18)
FactorBasisCreate(O,B): RngOrd, RngIntElt -> SeqEnum
EulerProduct(O, B) : RngOrd, RngIntElt -> FldReElt
AddRelation(E) : RngOrdElt -> BoolElt
EvaluateClassGroup(O) : RngOrd -> BoolElt
CompleteClassGroup(O) : RngOrd ->
FactorBasisVerify(O, L, U): RngOrd, RngIntElt, RngIntElt ->
ClassGroupSetUseMemory(O, f) : RngOrd, BoolElt ->
ClassGroupGetUseMemory(O) : RngOrd -> BoolElt
Setting the Class Group Bounds Globally
SetClassGroupBounds(n) : Any ->
SetClassGroupBoundMaps(f1, f2) : Map, Map ->
Example RngOrd_class-group-bounds (H34E19)
Unit Groups
UnitGroup(O) : RngOrd -> GrpAb, Map
UnitGroupAsSubgroup(O) : RngOrd -> GrpAb
TorsionUnitGroup(O) : RngOrd -> GrpAb, Map
IndependentUnits(O) : RngOrd -> GrpAb, Map
pFundamentalUnits(O, p) : RngOrd, RngIntElt -> GrpAb, Map
MergeUnits(K, a) : FldNum, FldNumElt -> BoolElt
UnitRank(O) : RngOrd -> RngIntElt
Example RngOrd_UnitGroup (H34E20)
IsExceptionalUnit(u) : RngOrdElt -> BoolElt
ExceptionalUnitOrbit(u) : RngOrdElt -> [ RngOrdElt ]
ExceptionalUnits(O) : RngOrd -> [ RngOrdElt ]
Solving 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 (H34E21)
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 (H34E22)
Unit Equations
UnitEquation(a, b, c) : FldNumElt, FldNumElt, FldNumElt -> [ ModHomElt ]
Example RngOrd_uniteq (H34E23)
Index Form Equations
IndexFormEquation(O, k) : RngOrd, RngIntElt -> [ RngOrdElt ]
Example RngOrd_index-form (H34E24)
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 (H34E25)
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
ResidueClassField(O, I) : RngOrd, RngOrdIdl -> FldFin, Map
Degree(I) : RngOrdIdl -> RngIntElt
Valuation(I, p) : RngOrdFracIdl, RngOrdIdl -> RngIntElt
Content(I) : RngOrdFracIdl -> RngIntElt
Example RngOrd_ideal-invar (H34E26)
Basis Representation
Basis(I) : RngOrdIdl -> [RngOrdElt]
BasisMatrix(I) : RngOrdFracIdl -> MtrxSpcElt
TransformationMatrix(I) : RngOrdFracIdl -> MtrxSpcElt, RngIntElt
CoefficientIdeals(I) : RngOrdFracIdl -> [RngOrdFracIdl]
Example RngOrd_ideal-basis (H34E27)
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 (H34E28)
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 / J : RngOrdFracIdl, RngOrdFracIdl -> RngOrdFracIdl
I div J : RngOrdIdl, RngOrdIdl -> RngOrdIdl
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) : RngOrdIdl, RngOrdIdl -> RngOrdIdl
IntegralSplit(I) : RngOrdFracIdl -> RngOrdIdl, RngElt
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>]
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
CoprimeRepresentative(I, J) : RngOrdIdl, RngOrdIdl -> FldOrdElt
ClassRepresentative(I) : RngOrdFracIdl -> RngOrdFracIdl
Lattice(I) : RngOrdIdl -> Lat, Map
Different(I) : RngOrdFracIdl -> RngOrdFracIdl
Codifferent(I) : RngOrdFracIdl -> RngOrdFracIdl
SUnitGroup(I) : RngOrdFracIdl -> GrpAb, Map
Example RngOrd_S-Units (H34E29)
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 (H34E30)
Quotient Rings
Operations on Quotient Rings
quo< O | I > : RngOrd, RngOrdIdl -> RngOrdRes
UnitGroup(OQ) : RngOrdRes -> GrpAb, Map
Modulus(OQ) : RngOrdRes -> RngOrdIdl
Example RngOrd_quotient (H34E31)
Elements of Quotients
OQ ! a : RngOrdRes, Elt -> 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 -> []
Reconstruction
ReconstructionEnvironment(p, k) : RngOrdIdl, RngIntElt -> RngOrdRecoEnv
Reconstruct(x, R) : RngOrdElt, RngOrdRecoEnv -> RngOrdElt
ChangePrecision(~ R, k) : RngOrdRecoEnv, RngIntElt ->
Example RngOrd_order-reco (H34E32)
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, p) : FldAlg, RngIntElt -> SeqEnum
Decomposition(K, p) : FldNum, PlcNumElt -> SeqEnum
Decomposition(m, p) :Map[FldRat, FldAlg], RngIntElt -> SeqEnum[<PlcNumElt, RngIntElt>]
InfinitePlaces(K) : FldAlg -> SeqEnum
Divisor(pl) : PlcNumElt -> DivNumElt
Divisor(I) : RngOrdFracIdl -> DivNumElt
Divisor(x) : FldNumElt -> DivNumElt
RealPlaces(K) : FldRat -> [PlcNumElt]
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
Evaluate(x, p) : FldOrdElt, PlcNumElt -> RngElt
RealEmbeddings(a) : RngOrdElt -> []
RealSigns(a) : RngOrdElt -> []
IsReal(p) : PlcNumElt -> BoolElt
IsComplex(p) : PlcNumElt -> BoolElt
IsFinite(p) : PlcNumElt -> BoolElt
IsInfinite(p) : PlcNumElt -> BoolElt, RngIntElt
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
Bibliography
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