Creation of General Algebraic Fields
NumberField(f) : RngUPolElt -> FldNum
NumberField(s) : [ RngUPolElt ] -> FldNum
ext< Q | s1, ..., sn > : FldRat, RngUPolElt, ..., RngUPolElt -> FldNum
Example RngOrd_Creation (H53E1)
RadicalExtension(F, d, a) : Rng, RngIntElt, RngElt -> FldAlg
SplittingField(F) : FldAlg -> FldAlg, SeqEnum
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 (H53E2)
OptimizedRepresentation(F) : FldAlg -> FldAlg, map
Example RngOrd_opt-rep (H53E3)
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
Example RngOrd_Orders (H53E4)
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 (H53E5)
OptimizedRepresentation(O) : RngOrd -> BoolElt, RngOrd, Map
O + P : RngOrd, RngOrd -> RngOrd
Order(O, T, d) : RngOrd, AlgMatElt, RngIntElt -> 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 (H53E6)
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
Example RngOrd_Elements (H53E7)
Creation of Homomorphisms
hom< F -> R | r > : FldAlg, Rng, RngElt -> Map
hom< O -> R | r > : RngOrd, Rng, RngElt -> Map
Example RngOrd_Homomorphisms (H53E8)
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 (H53E9)
Simplify(O) : RngOrd -> RngOrd
LLL(O) : RngOrd -> RngOrd, AlgMatElt
Example RngOrd_lll (H53E10)
Embed(F, L, a) : FldAlg, FldAlg, FldAlgElt ->
EmbeddingMap(F, L): FldAlg, FldAlg -> Map
Example RngOrd_em (H53E11)
Lattice(O) : RngOrd -> Lat, Map
MinkowskiSpace(F) : FldAlg -> Lat, Map
Completion(K, P) : FldAlg, RngOrdIdl -> FldLoc, Map
LocalRing(P, prec) : RngOrdIdl, RngIntElt -> RngLoc, Map
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 (H53E12)
Different(O) : RngOrd -> RngOrdIdl
Conductor(O) : RngOrd -> RngOrdIdl
Basis Representation
Basis(O) : RngOrd -> [ FldOrdElt ]
IntegralBasis(F) : FldAlg -> [ FldAlgElt ]
Example RngOrd_basis-ring (H53E13)
AbsoluteBasis(K) : FldAlg -> [FldAlgElt]
BasisMatrix(O) : RngOrd -> AlgMatElt
TransformationMatrix(O, P) : RngOrd -> AlgMatElt, RngIntElt
Example RngOrd_Bases (H53E14)
MultiplicationTable(O) : RngOrd -> [AlgMatElt]
TraceMatrix(O) : RngOrd -> AlgMatElt
Example RngOrd_MultiplicationTable (H53E15)
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
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
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 -> RngOrdElt
IsPower(a, k) : FldAlgElt, RngIntElt -> BoolElt, FldAlgElt
Denominator(a) : FldAlgElt -> RngIntElt
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
Real and Complex Valued Functions
AbsoluteValues(a) : FldAlgElt -> [FldPrElt]
AbsoluteLogarithmicHeight(a) : FldAlgElt -> FldPrElt
Conjugates(a) : FldAlgElt -> [ FldPrElt ]
Conjugate(a, k) : FldAlgElt, RngIntElt -> FldPrElt
Length(a) : FldAlgElt -> FldPrElt
Logs(a) : FldAlgElt -> [FldPrElt]
Example RngOrd_Discriminant (H53E16)
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
Other Functions
ElementToSequence(a) : FldAlgElt -> [ FldAlgElt ]
Flat(e) : FldAlgElt -> [ FldRatElt]
a[i] : FldAlgElt, RngIntElt -> FldRatElt
RepresentationMatrix(a) : FldAlgElt -> AlgMatElt
AbsoluteRepresentationMatrix(a) : FldAlgElt -> AlgMatElt
Example RngOrd_NormsEtc (H53E17)
Valuation(w, I) : RngOrdElt, RngOrdIdl -> RngIntElt
Decomposition(a): RngOrdElt -> SeqEnum[<RngOrdIdl, RngIntElt>]
Divisors(a) : RngOrdElt -> SeqEnum[RngOrdElt]
Index(a) : RngOrdElt -> RngIntElt
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 (H53E18)
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 (H53E19)
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 (H53E20)
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
FixedGroup(K, L) : FldAlg, FldAlg -> GrpPerm
DecompositionField(p) : RngOrdIdl -> FldNum, Map
RamificationField(p, i) : RngOrdIdl -> FldNum, Map
RamificationField(p) : RngOrdIdl -> FldNum, Map
InertiaField(p) : RngOrdIdl -> FldNum, Map
Example RngOrd_Ramification (H53E21)
Galois Groups
GaloisGroup(L) : FldAlg -> GrpPerm, [ FldPrElt, Any ]
Example RngOrd_GaloisGroups (H53E22)
Subfields
Subfields(K, n) : FldAlg -> [ < 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
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 (H53E23)
Ray Class Group
RayClassGroup(I) : RngOrdIdl -> GrpAb, Map
RayResidueRing(m) : RngOrdIdl -> GrpAb, Map
pSelmerGroup(O, p, S) : RngOrd P, prime p, { RngOrdIdl } -> G, m
Example RngOrd_ideal-ray (H53E24)
Abelian Extensions
RayClassField(m) : Map -> FldAb
EquationOrder(A) : FldAb -> RngOrd
Discriminant(A) : FldAb -> RngOrdIdl, [RngIntElt]
Conductor(A) : FldAb -> RngOrdIdl, [RngIntElt]
Components(A) : FldAb -> [RngOrd]
Example RngOrd_class-field (H53E25)
Norm Equations
NormEquation(O, m) : RngOrd, RngIntElt -> BoolElt, [ RngOrdElt ]
Example RngOrd_norm-equation (H53E26)
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 (H53E27)
Unit Equations
UnitEquation(a, b, c) : FldNumElt, FldNumElt, FldNumElt -> [ ModHomElt ]
Example RngOrd_uniteq (H53E28)
Index Form Equations
IndexFormEquation(O, k) : RngOrd, RngIntElt -> [ RngOrdElt ]
Example RngOrd_index-form (H53E29)
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 (H53E30)
Invariants
Order(I) : RngOrdFracIdl -> RngOrd
Denominator(I) : RngOrdFracIdl -> RngIntElt
PrimitiveElement(I) : RngOrdIdl -> RngOrdElt
Index(O, I) : RngOrd, RngOrdIdl -> RngIntElt
Norm(I) : RngOrdIdl -> RngIntElt
Minimum(I) : RngOrdFracIdl -> RngElt
AbsoluteNorm(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 (H53E31)
Basis Representation
Basis(I) : RngOrdIdl -> [RngOrdElt]
BasisMatrix(I) : RngOrdFracIdl -> MtrxSpcElt
TransformationMatrix(I) : RngOrdFracIdl -> MtrxSpcElt, RngIntElt
Example RngOrd_ideal-basis (H53E32)
Two--Element Presentations
Generators(I) : RngOrdIdl -> [ RngOrdElt ]
TwoElement(I) : RngOrdFracIdl -> FldOrdElt, FldOrdElt
TwoElementNormal(I) : RngOrdIdl -> RngOrdElt, RngOrdElt, RngIntElt
Example RngOrd_ideal-two (H53E33)
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
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
LCM(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
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>]
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
Lattice(I) : RngOrdIdl -> Lat, Map
SUnitGroup(I) : RngOrdFracIdl -> GrpAb, Map
Operations on Quotient Rings
quo< O | I > : RngOrd, RngOrdIdl -> RngOrdRes
UnitGroup(OQ) : RngOrdRes -> GrpAb, Map
Modulus(OQ) : RngOrdRes -> RngOrdIdl
Example RngOrd_quotient (H53E34)
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