Creation of General Algebraic Fields
NumberField(f) : RngUPolElt -> FldNum
NumberField(s) : [ RngUPolElt ] -> FldNum
ext< F | s1, ..., sn > : FldAlg, RngUPolElt, ..., RngUPolElt -> FldAlg
Example RngOrd_Creation (H48E1)
RadicalExtension(F, d, a) : Rng, RngIntElt, RngElt -> FldAlg
SplittingField(F) : FldAlg -> FldAlg, SeqEnum
SplittingField(f) : RngUPolElt -> FldAlg
SplittingField(L) : [RngUPolElt] -> FldNum, [FldNumElt]
sub< F | e_1, ..., e_n > : FldAlg, FldAlgElt, ..., FldAlgElt -> FldAlg, Map
MergeFields(F, L) : FldAlg, FldAlg -> SeqEnum
quo< FldNum : R | f > : RngUPol, RngUPolElt -> FldNum
Example RngOrd_CompositeFields (H48E2)
OptimizedRepresentation(F) : FldAlg -> FldAlg, map
Example RngOrd_opt-rep (H48E3)
Creation of Orders and Fields from Orders
MaximalOrder(F) : FldAlg -> RngOrd
MaximalOrder(f) : RngUPolElt -> RngOrd
MaximalOrder(O) : RngOrd -> RngOrd
EquationOrder(f) : RngUPolElt -> RngOrd
EquationOrder(K) : FldNum -> RngOrd
SubOrder(O) : RngOrd -> RngOrd
EquationOrder(O) : RngOrd -> RngOrd
Integers(O) : RngOrd -> RngOrd
Example RngOrd_Orders (H48E4)
sub< O | a_1, ..., a_r > : RngOrd, RngOrdElt, ..., RngOrdElt -> RngOrd
ext< O | a_1, ..., a_r > : 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 (H48E5)
OptimizedRepresentation(O) : RngOrd -> BoolElt, RngOrd, Map
O + P : RngOrd, RngOrd -> RngOrd
Order(O, T, d) : RngOrd, AlgMatElt, RngIntElt -> RngOrd
Order(O, M) : RngOrd, ModDed -> RngOrd
Order( [ e_1, ... e_n ] ): [FldAlgElt] -> RngOrd
Orders and Ideals
MultiplicatorRing(I) : RngOrdFracIdl -> Rng
pMaximalOrder(O, p) : RngOrd, RngIntElt -> RngOrd
pRadical(O, p) : RngOrd, RngIntElt -> RngOrdIdl
Example RngOrd_Round2 (H48E6)
Creation of Elements
F ! a : FldAlg, RngElt -> FldAlgElt
F ! [a_0, a_1, ..., a_(m - 1)] : FldAlg, [RngElt] -> FldAlgElt
O ! a : RngOrd, RngElt -> RngOrdElt
O ! [a_0, a_1, ..., a_(m - 1)] : RngOrd, [ RngElt ] -> RngOrdElt
Random(F, m) : FldAlg, RngIntElt -> FldAlgElt
Random(I, m) : RngOrdFracIdl, RngIntElt -> FldOrdElt
Example RngOrd_Elements (H48E7)
Creation of Homomorphisms
hom< F -> R | r > : FldAlg, Rng, RngElt -> Map
hom< O -> R | r > : RngOrd, Rng, RngElt -> Map
Example RngOrd_Homomorphisms (H48E8)
hom< O -> R | b_1, ..., b_n > : RngFunOrd, Rng, RngElt, ..., RngElt -> Map
IsRingHomomorphism(m) : Map -> BoolElt
Special Options
SetVerbose(s, n) : MonStgElt, RngIntElt ->
SetKantPrinting(f) : BoolElt -> BoolElt
SetKantPrecision(O, n) : RngOrd, RngIntElt ->
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 (H48E9)
Simplify(O) : RngOrd -> RngOrd
LLL(O) : RngOrd -> RngOrd, AlgMatElt
Example RngOrd_lll (H48E10)
Embed(F, L, a) : FldAlg, FldAlg, FldAlgElt ->
Embed(F, L, a) : FldAlg, FldAlg, [FldAlgElt] ->
EmbeddingMap(F, L): FldAlg, FldAlg -> Map
Example RngOrd_em (H48E11)
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 (H48E12)
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 (H48E13)
Different(O) : RngOrd -> RngOrdIdl
Conductor(O) : RngOrd -> RngOrdIdl
Basis Representation
Basis(O) : RngOrd -> [ FldOrdElt ]
IntegralBasis(F) : FldAlg -> [ FldAlgElt ]
Example RngOrd_basis-ring (H48E14)
AbsoluteBasis(K) : FldAlg -> [FldAlgElt]
BasisMatrix(O) : RngOrd -> AlgMatElt
TransformationMatrix(O, P) : RngOrd, RngOrd -> AlgMatElt, RngIntElt
Example RngOrd_Bases (H48E15)
MultiplicationTable(O) : RngOrd -> [AlgMatElt]
TraceMatrix(O) : RngOrd -> AlgMatElt
Example RngOrd_MultiplicationTable (H48E16)
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
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
IsAbsoluteField(K) : FldAlg -> BoolElt
IsWildlyRamified(K) : FldAlg -> BoolElt
IsTamelyRamified(K) : FldAlg -> BoolElt
IsUnramified(K) : FldAlg -> BoolElt
IsQuadratic(K) : FldAlg -> BoolElt, FldQuad
Setting Properties of Orders
SetOrderMaximal(O, b) : RngOrd, BoolElt ->
SetOrderTorsionUnit(O, e, r) : RngOrd, RngOrdElt, RngIntElt ->
SetOrderUnitsAreFundamental(O) : RngOrd ->
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
Predicates on Elements
IsIntegral(a) : FldAlgElt -> BoolElt
IsPrimitive(a) : FldAlgElt -> BoolElt
IsTorsionUnit(w) : RngOrdElt -> BoolElt
IsPower(w, n) : RngOrdElt, RngIntElt -> BoolElt, RngOrdElt
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 (H48E17)
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 (H48E18)
Other Functions
ElementToSequence(a) : FldAlgElt -> [ FldAlgElt ]
Eltseq(E, k) : FldAlgElt, Rng -> [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
ConditionalClassGroup(O) : RngOrd -> GrpAb, 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 (H48E19)
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 ->
Setting the Class Group Bounds Globally
SetClassGroupBounds(n) : RngIntElt ->
SetClassGroupBoundMaps(f1, f2) : Map, Map ->
Example RngOrd_class-group-bounds (H48E20)
Unit Groups
UnitGroup(O) : RngOrd -> GrpAb, Map
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 (H48E21)
IsExceptionalUnit(u) : RngOrdElt -> BoolElt
ExceptionalUnitOrbit(u) : RngOrdElt -> [ RngOrdElt ]
ExceptionalUnits(O) : RngOrd -> [ RngOrdElt ]
Automorphism Groups
Automorphisms(F) : FldAlg -> [ Map ]
AutomorphismGroup(F) : FldAlg -> GrpPerm, PowMap, Map
Example RngOrd_Automorphisms (H48E22)
AutomorphismGroup(K, F) : FldAlg, FldAlg -> GrpPerm, PowMap, Map
DecompositionGroup(p) : RngOrdIdl -> GrpPerm
RamificationGroup(p, i) : RngOrdIdl, RngIntElt -> GrpPerm
RamificationGroup(p) : RngOrdIdl -> GrpPerm
InertiaGroup(p) : RngOrdIdl -> GrpPerm
FixedField(K, U) : FldAlg, GrpPerm -> FldNum, Map
FixedField(K, S) : FldAlg, [Map] -> FldNum, Map
FixedGroup(K, L) : FldAlg, FldAlg -> GrpPerm
DecompositionField(p) : RngOrdIdl -> FldNum, Map
RamificationField(p, i) : RngOrdIdl, RngIntElt -> FldNum, Map
RamificationField(p) : RngOrdIdl -> FldNum, Map
InertiaField(p) : RngOrdIdl -> FldNum, Map
Example RngOrd_Ramification (H48E23)
Galois Groups
GaloisGroup(L) : FldAlg[FldAlg] -> GrpPerm, [ FldPrElt ], Any
GaloisGroup(f) : RngUPolElt[RngIntElt] -> GrpPerm, SeqEnum, GaloisData
GaloisGroup(K) : FldAlg[FldRat] -> GrpPerm, SeqEnum, GaloisData
GaloisProof(f, S) : RngUPolElt, GaloisData -> BoolElt
GaloisRoot(f, i, S) : RngUPolElt, RngIntElt, GaloisData -> RngElt
GaloisRoot(i, S) : RngIntElt, GaloisData -> RngElt
Stauduhar(G, H, S, B) : GrpPerm, GrpPerm, GaloisData, RngIntElt -> RngIntElt, GrpPermElt, BoolElt, UserProgram
Example RngOrd_GaloisGroups (H48E24)
GaloisSubgroup(K, U) : FldNum, GrpPerm -> FldNum, UserProgram
GaloisQuotient(K, Q) : FldNum, GrpPerm -> SeqEnum[FldNum]
IsInt(x, B, S) : RngElt, RngIntElt, GaloisData -> BoolElt, RngElt
Subfields
Subfields(K, n) : FldAlg, RngIntElt -> [ < FldAlg, Hom > ]
Subfields(K) : FldAlg -> [ < FldAlg, Hom > ]
The Subfield Lattice
SubfieldLattice(K) : FldNum -> SubFldLat
# L : SubFldLat -> RngIntElt
Bottom(L) : SubFldLat -> SubFldLatElt
Top(L) : SubFldLat -> SubFldLatElt
Random(L) : SubFldLat -> SubFldLatElt
L ! n : SubFldLat, RngIntElt -> SubFldLatElt
NumberField(e) : SubFldLatElt -> FldNum
EmbeddingMap(e) : SubFldLatElt -> Map
Degree(e) : SubFldLatElt -> RngIntElt
e eq f : SubFldLatElt, SubFldLatElt -> BoolElt
e subset f : SubFldLatElt, SubFldLatElt -> BoolElt
e * f : SubFldLatElt, SubFldLatElt -> SubFldLatElt
e meet f : SubFldLatElt, SubFldLatElt -> SubFldLatElt
&meet S : [ SubFldLatElt ] -> SubFldLatElt
MaximalSubfields(e) : SubFldLatElt -> [ SubFldLatElt ]
MinimalOverfields(e) : SubFldLatElt -> [ SubFldLatElt ]
Example RngOrd_SubfieldLattice (H48E25)
Norm Equations
NormEquation(O, m) : RngOrd, RngIntElt -> BoolElt, [ RngOrdElt ]
NormEquation(F, m) : FldAlg, RngIntElt -> BoolElt, [ FldAlgElt ]
NormEquation(m, N): RngElt, Map -> BoolElt, RngElt
Example RngOrd_norm-equation (H48E26)
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 (H48E27)
Unit Equations
UnitEquation(a, b, c) : FldNumElt, FldNumElt, FldNumElt -> [ ModHomElt ]
Example RngOrd_uniteq (H48E28)
Index Form Equations
IndexFormEquation(O, k) : RngOrd, RngIntElt -> [ RngOrdElt ]
Example RngOrd_index-form (H48E29)
Creation of Ideals in Orders
x * O : RngElt, RngOrd -> RngOrdFracIdl
F !! I : FldOrd, RngOrdFracIdl -> RngOrdFracIdl
ideal< O | a_1, a_2, ... , a_m > : RngOrd, RngElt, ..., RngElt -> RngOrdFracIdl
Example RngOrd_Ideals (H48E30)
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) : 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 (H48E31)
Basis Representation
Basis(I) : RngOrdIdl -> [RngOrdElt]
BasisMatrix(I) : RngOrdFracIdl -> MtrxSpcElt
TransformationMatrix(I) : RngOrdFracIdl -> MtrxSpcElt, RngIntElt
Example RngOrd_ideal-basis (H48E32)
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 (H48E33)
Predicates on Ideals
IsIntegral(I) : RngOrdFracIdl -> BoolElt
IsZero(I) : RngOrdFracIdl -> 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
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
I meet J : RngOrdFracIdl, 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>]
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
SUnitGroup(I) : RngOrdFracIdl -> GrpAb, Map
Example RngOrd_S-Units (H48E34)
SUnitAction(SU, Act, S) : Map, Map, SeqEnum[RngOrdIdl] -> Map
SUnitDiscLog(SU, x, S) : Map, FldAlgElt, SeqEnum[RngOrdIdl] -> GrpAbElt
Example RngOrd_S-Units, advanced (H48E35)
Different(I) : RngOrdFracIdl -> RngOrdFracIdl
Operations on Quotient Rings
quo< O | I > : RngOrd, RngOrdIdl -> RngOrdRes
UnitGroup(OQ) : RngOrdRes -> GrpAb, Map
Modulus(OQ) : RngOrdRes -> RngOrdIdl
Example RngOrd_quotient (H48E36)
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 -> []
Creation of Structures
Places(K) : FldNum -> PlcNum
Creation of Elements
Place(I) : RngOrdIdl -> PlcNumElt
Decomposition(K, p) : FldNum, RngIntElt -> SeqEnum
Decomposition(K, p) : FldNum, PlcNumElt -> SeqEnum
InfinitePlaces(K) : FldAlg -> SeqEnum
Divisor(pl) : PlcNumElt -> DivNumElt
Divisor(I) : RngOrdFracIdl -> DivNumElt
Divisor(x) : FldNumElt -> DivNumElt
Arithmetic with Places and Divisors
Other functions for Divisors and Places
Valuation(a, p) : FldNumElt, PlcNumElt -> RngElt
Valuation(I, p) : RngOrdFracIdl , PlcNumElt -> RngElt
Support(D) : DivNumElt -> SeqEnum, SeqEnum
Ideal(D) : DivNumElt -> RngOrdIdl
Evaluate(x, p) : FldNumElt, PlcNumElt -> RngElt
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