NUMBER FIELDS

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NUMBER FIELDS

Acknowledgements

Introduction

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

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

Class and Unit Groups

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

Characters
Creation Functions
Functions on Groups and Group Elements
Predicates on Group Elements
Passing between Dirichlet and Hecke Characters
L-functions of Hecke Characters
Hecke Grössencharacters and their L-functions

Number Field Database
Creation
Access

Bibliography

DETAILS

Introduction
Example FldNum_Q-as-number-field (H34E1)

Creation Functions

Creation of Number Fields
NumberField(f) : RngUPolElt -> FldNum
RationalsAsNumberField() : FldRat -> FldNum
NumberField(s) : [ RngUPolElt ] -> FldNum
ext< F | s1, ..., sn > : FldNum, RngUPolElt, ..., RngUPolElt -> FldNum
Example FldNum_Creation (H34E2)
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, ..., e_n > : 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 (H34E3)
OptimizedRepresentation(F) : FldNum -> FldNum, map
Example FldNum_opt-rep (H34E4)

Maximal Orders
MaximalOrder(F) : FldNum -> RngOrd

Creation of Elements
F ! a : FldNum, RngElt -> FldNumElt
F ! [a₀, a₁, ..., a_{m - 1}] : FldNum, [RngElt] -> FldNumElt
Random(F, m) : FldNum, RngIntElt -> FldNumElt
Example FldNum_Elements (H34E5)

Creation of Homomorphisms
hom< F -> R | r > : FldNum, Rng, RngElt -> Map
Example FldNum_Homomorphisms (H34E6)

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
Example FldNum_Compositum (H34E7)
Embed(F, L, a) : FldNum, FldNum, FldNumElt ->
Embed(F, L, a) : FldNum, FldNum, [FldNumElt] ->
EmbeddingMap(F, L): FldNum, FldNum -> Map
Example FldNum_em (H34E8)
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 (H34E9)

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 (H34E10)

Basis Representation
Basis(F) : FldNum -> [ FldNumElt ]
IntegralBasis(F) : FldNum -> [ FldNumElt ]
Example FldNum_basis-ring (H34E11)
AbsoluteBasis(K) : FldNum -> [FldNumElt]
Example FldNum_Bases (H34E12)

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

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

Finding Special Elements
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 Valued Functions
AbsoluteValues(a) : FldNumElt -> [FldComElt]
AbsoluteLogarithmicHeight(a) : FldNumElt -> FldComElt
Conjugates(a) : FldNumElt -> [ FldComElt ]
Conjugate(a, k) : FldNumElt, RngIntElt -> FldComElt
Conjugate(a, l) : FldNumElt, [RngIntElt] -> FldComElt
Length(a) : FldNumElt -> FldReElt
Logs(a) : FldNumElt -> [FldReElt]
CoefficientHeight(E) : FldNumElt -> RngIntElt
CoefficientLength(E) : FldNumElt -> RngIntElt

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 (H34E13)

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 and Unit Groups
ClassGroup(K: parameters) : FldAlg -> GrpAb, Map
ConditionalClassGroup(K) : FldAlg -> GrpAb, Map
ClassNumber(K: parameters) : FldAlg -> RngIntElt
BachBound(K) : FldNum -> RngIntElt
MinkowskiBound(K) : FldNum -> RngIntElt
UnitGroup(K) : FldNum -> GrpAb, Map
TorsionUnitGroup(K) : FldNum -> GrpAb, Map
UnitRank(K) : FldNum -> RngIntElt
Example FldNum_UnitGroup (H34E14)

Galois Theory
GaloisGroup(K) : FldNum -> GrpPerm, [RngElt], GaloisData

Solving Norm Equations
NormEquation(F, m) : FldNum, RngIntElt -> BoolElt, [ FldNumElt ]
NormEquation(m, N): RngElt, Map -> BoolElt, RngElt
SimNEQ(K, e, f) : FldNum, FldNumElt, FldNumElt -> BoolElt, [FldNumElt]
Example FldNum_norm-equation (H34E15)

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

Creation Functions
DirichletGroup(I) : RngOrdIdl -> GrpDrchNF
HeckeCharacterGroup(I) : RngOrdIdl -> GrpHecke
UnitTrivialSubgroup(G) : GrpDrchNF -> GrpDrchNF
TotallyUnitTrivialSubgroup(G) : GrpDrchNF -> GrpDrchNF

Functions on Groups and Group Elements
Modulus(G) : GrpDrchNF -> RngOrdIdl, SeqEnum
Order(chi) : GrpDrchNFElt -> RngIntElt
Random(G) : GrpDrchNF -> GrpDrchNFElt
Domain(G) : GrpDrchNF -> FldNum
Domain(G) : GrpHecke -> PowIdl
Decomposition(chi) : GrpDrchNFElt -> List
Conductor(chi) : GrpDrchNFElt -> RngOrdIdl, SeqEnum
AssociatedPrimitiveCharacter(chi) : GrpDrchNFElt -> GrpDrchNFElt
Restrict(chi, D) : GrpDrchNFElt, GrpDrchNF -> GrpDrchNFElt
TargetRestriction(G, C) : GrpDrchNF, FldCyc -> GrpDrchNF
SetTargetRing(~chi, e) : GrpDrchNFElt, RngElt ->
Extend(chi, D) : GrpDrchNFElt, GrpDrchNF -> GrpDrchNFElt, GrpDrchNF

Predicates on Group Elements
IsTrivial(chi) : GrpDrchNFElt -> BoolElt
IsTrivialOnUnits(chi) : GrpDrchNFElt -> BoolElt
IsOdd(chi) : GrpDrchNFElt -> BoolElt
IsEven(chi) : GrpDrchNFElt -> BoolElt
IsTotallyEven(chi) : GrpDrchNFElt -> BoolElt
IsPrimitive(chi) : GrpDrchNFElt -> BoolElt

Passing between Dirichlet and Hecke Characters
HeckeLift(chi) : GrpDrchNFElt -> GrpHeckeElt, GrpHecke
DirichletRestriction(psi) : GrpHeckeElt -> GrpDrchNFElt
NormInduction(K, chi) : FldNum, GrpDrchElt -> GrpHeckeElt
Example FldNum_dirichletQ (H34E16)
DirichletCharacter(I, B) : RngOrdIdl, Tup -> GrpDrchNFElt, GrpDrchNF
Example FldNum_dirichlet-hecke (H34E17)
CentralCharacter(chi) : GrpDrchNFElt -> GrpDrchNFElt
Example FldNum_central-chars (H34E18)
DirichletCharacterOverNF(chi) : GrpDrchElt -> GrpDrchNFElt
Example FldNum_dirich-chars-over-nf-and-q (H34E19)

L-functions of Hecke Characters
Example FldNum_lfunc-hecke (H34E20)

Hecke Grössencharacters and their L-functions
Grossencharacter(psi, chi, T) : GrpHeckeElt, GrpDrchNFElt, SeqEnum -> GrossenChar
Grossencharacter(psi, T) : GrpHeckeElt, SeqEnum -> GrossenChar
Conductor(psi) : GrossenChar -> RngOrdIdl, SeqEnum
Extend(psi, I) : GrossenChar, RngOrdIdl -> GrossenChar
CentralCharacter(psi) : GrossenChar -> GrpDrchNFElt
GrossenTwist(Y, D) : GrossenChar, List -> GrossenChar
Example FldNum_grossenchar-gaussian (H34E21)
Example FldNum_grossenchar-sqrt23 (H34E22)
Example FldNum_grossenchar-symcubed-sqrt59 (H34E23)
Example FldNum_grossen-char-cyclo5 (H34E24)
Example FldNum_grossenchar-embedding (H34E25)
Example FldNum_grossen-large-gamma (H34E26)
Example FldNum_grossen-cyclo8 (H34E27)
Example FldNum_hypgeom-mot (H34E28)

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 (H34E29)
Example FldNum_anfdb-basic2 (H34E30)

Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013