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NUMBER FIELDS
Acknowledgements
Introduction
Acknowledgement
Creation Functions
Creation of Number Fields
Maximal Orders
Creation of Elements
Creation of Homomorphisms
Structure Operations
General Functions
Related Structures
Representing Fields as Vector Spaces
Invariants
Basis Representation
Ring Predicates
Field Predicates
Fields with a Labelled Embedding
Creation Functions
Composition and Intersection
Embedding and Reconstruction
Element Operations
Parent and Category
Arithmetic
Equality and Membership
Predicates on Elements
Field Generators
Real and Complex Embeddings
Heights
Norm, Trace, and Minimal Polynomial
Other Functions
Class Group and Unit Group
Galois Theory
Solving Norm Equations
Places and Divisors
Creation of Structures
Operations on Structures
Creation of Elements
Arithmetic with Places and Divisors
Other Functions for Places and Divisors
Number Field Database
Creation
Access
Bibliography
Introduction
Acknowledgement
Creation Functions
Creation of Number Fields
NumberField(f) : RngUPolElt -> FldNum
RationalsAsNumberField() : -> FldNum
NumberField(s) : [ RngUPolElt ] -> FldNum
ext< F | s1, ..., sn > : FldNum, RngUPolElt, ..., RngUPolElt -> FldNum
Example FldNum_Creation (H36E1)
RadicalExtension(F, d, a) : Rng, RngIntElt, RngElt -> FldNum
SplittingField(F) : FldNum -> FldNum, SeqEnum
SplittingField(f) : RngUPolElt -> FldNum
SplittingField(L) : [RngUPolElt] -> FldNum, [FldNumElt]
sub< F | e1, ..., en > : FldAlg, FldAlgElt, ..., FldAlgElt -> FldAlg, Map
MergeFields(F, L) : FldNum, FldNum -> SeqEnum
Compositum(K, L) : FldNum, FldNum -> FldNum
quo< FldNum : R | f > : RngUPol, RngUPolElt -> FldNum
Example FldNum_CompositeFields (H36E2)
OptimizedRepresentation(F) : FldNum -> FldNum, Map
Example FldNum_opt-rep (H36E3)
Maximal Orders
MaximalOrder(F) : FldNum -> RngOrd
Creation of Elements
F ! a : FldNum, RngElt -> FldNumElt
F ! [a0, a1, ..., am - 1] : FldNum, [RngElt] -> FldNumElt
Random(F, m) : FldNum, RngIntElt -> FldNumElt
Example FldNum_Elements (H36E4)
Creation of Homomorphisms
hom< F -> R | r > : FldNum, Rng, RngElt -> Map
Example FldNum_Homomorphisms (H36E5)
Structure Operations
General Functions
AssignNames(~K, s) : FldNum, [ MonStgElt ] ->
Name(K, i) : FldNum, RngIntElt -> FldNumElt
Related Structures
GroundField(F) : FldNum -> Fld
AbsoluteField(F) : FldNum -> FldNum
SimpleExtension(F) : FldNum -> FldNum
RelativeField(F, L) : FldNum, FldNum -> FldNum
Components(F) : FldNum -> [FldNum]
Example FldNum_Compositum (H36E6)
Embed(F, L, a) : FldNum, FldNum, FldNumElt ->
Embed(F, L, a) : FldNum, FldNum, [FldNumElt] ->
EmbeddingMap(F, L): FldNum, FldNum -> Map
Example FldNum_em (H36E7)
MinkowskiSpace(F) : FldNum -> Lat, Map
Completion(K, P) : FldNum, RngOrdIdl -> FldLoc, Map
Completion(K, P) : FldNum, PlcNumElt -> FldLoc, Map
Representing Fields as Vector Spaces
Algebra(K, J) : FldNum, Fld -> AlgAss, Map
VectorSpace(K, J) : FldNum, Fld -> ModTupFld, Map
Example FldNum_vector_space_eg (H36E8)
Invariants
Degree(F) : FldNum -> RngIntElt
AbsoluteDegree(F) : FldNum -> RngIntElt
Discriminant(F) : FldNum -> RngIntElt
AbsoluteDiscriminant(K) : FldNum -> FldRatElt
Regulator(K) : FldNum -> FldComElt
RegulatorLowerBound(K) : FldNum -> FldComElt
Signature(F) : FldAlg -> RngIntElt, RngIntElt
UnitRank(K) : FldNum -> RngIntElt
DefiningPolynomial(F) : FldNum -> RngUPolElt
Zeroes(F, n) : FldNum, RngIntElt -> [ FldComElt ]
Example FldNum_zero (H36E9)
Basis Representation
Basis(F) : FldNum -> [ FldNumElt ]
IntegralBasis(F) : FldNum -> [ FldNumElt ]
Example FldNum_basis-ring (H36E10)
AbsoluteBasis(K) : FldNum -> [FldNumElt]
Example FldNum_Bases (H36E11)
Ring Predicates
F eq L : FldNum, FldNum -> BoolElt
IsEuclideanDomain(F) : FldNum -> BoolElt
IsSimple(F) : FldNum -> BoolElt
IsPrincipalIdealRing(F) : FldNum -> BoolElt
HasComplexConjugate(K) : FldNum -> BoolElt, Map
ComplexConjugate(x) : FldNumElt -> FldNumElt
Field Predicates
IsIsomorphic(F, L) : FldNum, FldNum -> BoolElt, Map
IsSubfield(F, L) : FldNum, FldNum -> BoolElt, Map
IsNormal(F) : FldNum -> BoolElt
IsAbelian(F) : FldNum -> BoolElt
IsCyclic(F) : FldNum -> BoolElt
IsAbsoluteField(K) : FldNum -> BoolElt
Fields with a Labelled Embedding
Creation Functions
EmbeddedNumberField(L, c) : FldNum, FldComElt -> BoolElt, FldNumEmb
EmbeddedNumberField(f, r) : RngUPolElt, FldReElt -> BoolElt, FldNumEmb
EmbeddedNumberField(f, i) : RngUPolElt, RngIntElt -> FldNumEmb
EmbeddedNumberField(L, i) : FldNum, RngIntElt -> FldNumEmb
EmbeddedSplittingField(f) : RngUPolElt -> FldNumEmb, SeqEnum
Subfield(L, K) : FldNumEmb, FldNum -> FldNumEmb
Composition and Intersection
Composite(K, L) : FldNumEmb, FldNumEmb -> FldNumEmb
Intersection(K, L) : FldNumEmb, FldNumEmb -> FldNumEmb
IsSubfieldEmb(K, L) : FldNumEmb, FldNumEmb -> BoolElt
Embedding and Reconstruction
ComplexImage(x) : FldNumElt -> FldComElt
Embedding(K) : FldNumEmb -> UserProgram
Reconstruction(L, x) : FldNumEmb, FldComElt -> BoolElt, FldNumElt
Example FldNum_fldemb-eg-1 (H36E12)
Element Operations
Parent and Category
Arithmetic
Sqrt(a) : FldNumElt -> FldNumElt
Root(a, n) : FldNumElt, RngIntElt -> FldNumElt
IsPower(a, k) : FldNumElt, RngIntElt -> BoolElt, FldNumElt
Denominator(a) : FldNumElt -> RngIntElt
Numerator(a) : FldNumElt -> RngIntElt
Qround(E, M): FldNumElt, RngIntElt -> FldNumElt
Equality and Membership
Predicates on Elements
IsIntegral(a) : FldNumElt -> BoolElt, RngIntElt
IsPrimitive(a) : FldNumElt -> BoolElt
IsTotallyPositive(a) : FldNumElt -> BoolElt
Field Generators
K . 1 : FldNum -> FldNumElt
PrimitiveElement(K) : FldNum -> FldNumElt
Generators(K): FldNum -> FldNumElt
GeneratorsOverBaseRing(K) : FldNum -> FldNumElt
GeneratorsSequence(K): FldNum -> [FldNumElt]
GeneratorsSequenceOverBaseRing(K) : FldNum -> [FldNumElt]
Generators(K, k) : FldAlg, FldAlg -> [FldAlgElt]
Real and Complex Embeddings
Heights
Norm, Trace, and Minimal Polynomial
Norm(a) : FldNumElt -> FldNumElt
AbsoluteNorm(a) : FldNumElt -> FldRatElt
Trace(a) : FldNumElt -> FldNumElt
AbsoluteTrace(a) : FldNumElt -> FldRatElt
CharacteristicPolynomial(a) : FldNumElt -> RngUPolElt
AbsoluteCharacteristicPolynomial(a) : FldNumElt -> RngUPolElt
MinimalPolynomial(a) : FldNumElt -> RngUPolElt
AbsoluteMinimalPolynomial(a) : FldNumElt -> RngUPolElt
RepresentationMatrix(a) : FldNumElt -> NumMatElt
AbsoluteRepresentationMatrix(a) : FldNumElt -> NumMatElt
Example FldNum_NormsEtc (H36E13)
Other Functions
ElementToSequence(a) : FldNumElt -> [ FldNumElt ]
Eltseq(E, k) : FldNumElt, FldNum -> [RngElt]
Flat(e) : FldNumElt -> [ FldRatElt]
a[i] : FldNumElt, RngIntElt -> FldRatElt
ProductRepresentation(a) : FldNumElt -> [ FldNumElt ], [ RngIntElt ]
ProductRepresentation(P, E) : [ FldNumElt ], [ RngIntElt ] -> FldNumElt
Class Group and Unit Group
Example FldNum_ClassGroupUnitGroup (H36E14)
Galois Theory
GaloisGroup(K) : FldNum -> GrpPerm, [RngElt], GaloisData
Solving Norm Equations
Example FldNum_norm-equation (H36E15)
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) : FldNum, RngIntElt -> SeqEnum
Decomposition(K, p) : FldNum, PlcNumElt -> SeqEnum
Decomposition(m, p) : Map[FldRat, FldNum], RngIntElt -> SeqEnum[<PlcNumElt, RngIntElt>]
InfinitePlaces(K) : FldNum -> 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) : FldNumElt, PlcNumElt -> RngElt
RealEmbeddings(a) : FldNumElt -> []
RealSigns(a) : FldNumElt -> []
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
DecompositionGroup(P) : PlcNumElt -> GrpPerm
Number Field Database
Creation
NumberFieldDatabase(d) : RngIntElt -> DB
sub< D | dmin, dmax : parameters> : DB, RngIntElt, RngIntElt -> DB
Access
Degree(D) : DB -> RngIntElt
DiscriminantRange(D) : DB -> RngIntElt, RngIntElt
# D : DB -> RngIntElt
NumberOfFields(D, d) : DB, RngIntElt -> RngIntElt
NumberFields(D) : DB -> [ FldNum ]
NumberFields(D, d) : DB, RngIntElt -> [ FldNum ]
Example FldNum_anfdb-basic1 (H36E16)
Example FldNum_anfdb-basic2 (H36E17)
Bibliography
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