15 Subfields of Cyclotomic Fields

The only number fields that GAP3 can handle at the moment are subfields of cyclotomic fields, e.g., Q(√5) is a number field that is not cyclotomic but contained in the cyclotomic field Q5 = Q(e(2π i)/(5)). Although this means that GAP3 does not know arbitrary algebraic number fields but only those with abelian Galois group, here we call these fields number fields for short. The elements of number fields are called cyclotomics (see chapter Cyclotomics). Thus number fields are the domains (see chapter Domains) related to cyclotomics; they are special field records (see Field Records) which are needed to specify the field extension with respect to which e.g. the trace of a cyclotomic shall be computed.

In many situations cyclotomic fields need not be treated in a special way, except that there may be more efficient algorithms for them than for arbitrary number fields. For that, there are the global variables NumberFieldOps and CyclotomicFieldOps, both records which contain the field operations stored in FieldOps (see chapter Fields) and in which some functions are overlaid (see Domain Functions for Number Fields). If all necessary information about a function is already given in chapter Fields, this function is not described here; this is the case e.g. for Conjugates and related functions, like Trace and CharPol. Some functions, however, need further explanation, e.g., Coefficients for Number Fields tells more about Coefficients for number fields.

There are some functions which are different for cyclotomic fields and other number fields, e.g., the field constructors CF resp. NF. In such a situation, the special case is described in a section immediately following the section about the general case.

Besides the single number fields, there is another domain in GAP3 related to number fields, the domain Cyclotomics of all cyclotomics. Although this is an abstract field, namely the field Qab, Cyclotomics is not a field record. It is used by DefaultField, DefaultRing, Domain, Field and Ring (see DefaultField, DefaultRing, Domain, Field, Ring) which are mainly interested in the corresponding entries of Cyclotomics.operations since these functions know how to create fields resp. integral rings generated by some cyclotomics.

characteristic functions (see IsNumberField, IsCyclotomicField),
field constructors (see Number Field Records, Cyclotomic Field Records),
(default) fields of cyclotomics (see DefaultField and Field for Cyclotomics), and (default) rings of cyclotomic integers (see DefaultRing and Ring for Cyclotomic Integers),
Galois groups of number fields (see GeneratorsPrimeResidues, GaloisGroup for Number Fields),
vector space bases (see ZumbroichBase, Integral Bases for Number Fields, NormalBaseNumberField) and coefficients (see Coefficients for Number Fields) and
overlaid functions in the operations records (see Domain Functions for Number Fields).

The external functions are in the file LIBNAME/"numfield.g"

15.1 IsNumberField

IsNumberField( obj )

returns true if obj is a field record (see IsField, Field Records) of a field of characteristic zero where F.generators is a list of cyclotomics (see chapter Cyclotomics), and false else.

    gap> IsNumberField( CF(9) ); IsNumberField( NF( [ ER(3) ] ) );
true
true
gap> IsNumberField( GF( 2 ) );
false

15.2 IsCyclotomicField

IsCyclotomicField( obj )

returns true if obj is a number field record (see IsNumberField) where obj.isCyclotomicField = true, and false else.

    gap> IsCyclotomicField( CF(9) );
true
gap> IsCyclotomicField( NF( [ ER(-3) ] ) );
true
gap> IsCyclotomicField( NF( [ ER(3) ] ) );
false

15.3 Number Field Records

NumberField( gens )
NumberField( n, stab )
NumberField( subfield, poly )
NumberField( subfield, base )

NumberField may be abbreviated NF; it returns number fields, namely

NumberField( gens ):
the number field generated by the cyclotomics in the list gens,

NumberField( n, stab ):
the fixed field of the prime residues in the list stab inside the cyclotomic field Qn (see Cyclotomic Field Records),

NumberField( subfield, poly ):
the splitting field of the polynomial poly (which must have degree at most 2) over the number field subfield; subfield = 0 is equivalent to subfield = Rationals,

NumberField( subfield, base ):
the extension field of the number field subfield which is as vector space generated by the elements of the list base of cyclotomics; that means, base must be or at least contain a vector space base of this extension, if base is a base it will be assigned to the base field of the cyclotomic field (see Coefficients for Number Fields). subfield = 0 is equivalent to subfield = Rationals.

    gap> NF( [ EB(7), ER(3) ] );
NF(84,[ 1, 11, 23, 25, 37, 71 ])
gap> NF( 7, [ 1 ] );
CF(7)
gap> NF( NF( [ EB(7) ] ), [ 1, 1, 1 ] );
NF(NF(7,[ 1, 2, 4 ]),[ 1, E(3) ])
gap> F:= NF( 0, [ 1, E(4) ] ); G:= NF( 0, NormalBaseNumberField( F ) );
GaussianRationals
CF( Rationals,[ 1/2-1/2*E(4), 1/2+1/2*E(4) ])
gap> G.base; G.basechangemat; Coefficients( G, 1 );
[ 1/2-1/2*E(4), 1/2+1/2*E(4) ]
[ [ 1, 1 ], [ -1, 1 ] ]
[ 1, 1 ]

Number field records are field records (see Field Records) representing a number field. Besides the obligatory record components, a number field record F contains the component

stabilizer:

the list of prime residues modulo NofCyc( F.generators ) which fix all elements of F

and possibly

isIntegralBase:

true if F.base is an integral vector space base of the field extension F / F.field, false else (used by Ring Ring); for the case that F.field is a cyclotomic field, Integral Bases for Number Fields describes integral bases of the field extension;

isNormalBase:

true if F.base is a normal vector space base of the field extension F/F.field, false else;

coeffslist:

a list of integers used by Coefficients Coefficients; (see also Coefficients for Number Fields);

coeffsmat:

a matrix of cyclotomics used by Coefficients Coefficients; bound only if F.field is not a cyclotomic field (see also Coefficients for Number Fields);

basechangemat:

square matrix of dimension F.dimension, representing the basechange from the default base of F / F.field (see Coefficients for Number Fields) to the base stored in F.base if these two are different; used by Coefficients.

Note: These fields and also the field base should not be changed by hand!

15.4 Cyclotomic Field Records

CyclotomicField( n )
CyclotomicField( gens )
CyclotomicField( subfield, n )
CyclotomicField( subfield, base )

CyclotomicField may be abbreviated CF; it returns cyclotomic fields, namely

CyclotomicField( n ):

the field Qn (over the rationals),

CyclotomicField( gens ):

the smallest cyclotomic field containing the cyclotomics in the list gens (over the rationals),

CyclotomicField( subfield, n ):

the field Qn over the number field subfield,

CyclotomicField( subfield, base ):

the cyclotomic extension field of the number field subfield which is as vector space generated by the elements of the list base of cyclotomics; that means, base must be or at least contain a vector space base of this extension, if base is a base it will be assigned to the base field of the cyclotomic field (see Coefficients for Number Fields). subfield = 0 is equivalent to subfield = Rationals.

    gap> CF( 5 ); CF( [ EB(7), ER(3) ] ); CF( NF( [ ER(3) ] ), 24 );
CF(5)
CF(84)
CF(24)/NF(12,[ 1, 11 ])
gap> CF( CF(3), [ 1, E(4) ] );
CF(12)/CF(3)

A cyclotomic field record is a field record (see Field Records), in particular a number field record (see Number Field Records) that represents a cyclotomic field. Besides the obligatory record fields, a cyclotomic field record F contains the fields

isCyclotomicField:

always true; used by IsCyclotomicField IsCyclotomicField,

zumbroichbase:

a list containing ZumbroichBase( n, m ) (see ZumbroichBase) if F represents the field extension Qn/Qm, and containing Zumbroichbase( n, 1 ) if F is an extension of a number field that is not cyclotomic; used by Coefficients Coefficients, see Coefficients for Number Fields

and possibly optional fields of number fields (see Number Field Records).

15.5 DefaultField and Field for Cyclotomics

For a set S of cyclotomics,

DefaultField( S ) = CF( S ) is the smallest cylotomic field containing S (see DefaultField), the so--called conductor of S;

Field( S ) = NF( S ) is the smallest field containing S (see Field).

    gap> DefaultField( [ E(5) ] ); DefaultField( [ E(3), ER(6) ] );
CF(5)
CF(24)
gap> Field( [ E(5) ] ); Field( [ E(3), ER(6) ] );
CF(5)
NF(24,[ 1, 19 ])

DefaultField and Field are used by functions that specify the field for which some cyclotomics are regarded as elements (see DefaultField, Field), e.g., Trace with only one argument will compute the trace of this argument (which must be a cyclotomic) with respect to its default field.

15.6 DefaultRing and Ring for Cyclotomic Integers

For a set S of cyclotomic integers,

DefaultRing( S ) is the ring of integers in CF( S ) (see DefaultRing),

Ring( S ) is the ring of integers in NF( S ) (see Ring).

    gap> Ring( [ E(5) ] );
Ring( E(5) )
gap> Ring( [ EB(7) ] );
Ring( E(7)+E(7)^2+E(7)^4 )
gap> DefaultRing( [ EB(7) ] );
Ring( E(7) ) 

15.7 GeneratorsPrimeResidues

GeneratorsPrimeResidues( n )

returns a record with fields

primes:

the set of prime divisors of the integer n,

exponents:

the corresponding exponents in the factorization of n and

generators:

generators of the group of prime residues: For each odd prime p there is one generator, corresponding to a primitive root of the subgroup (Z/pνp)* of (Z/nZ)*, where νp is the exponent of p in the factorization of n; for p = 2, we have one generator in the case that 8 does not divide n, and a list of two generators (corresponding to ⟨* 5, *(2ν2-1)⟩ = (Z/2ν2)*) else.

    gap> GeneratorsPrimeResidues( 9 );      # 2 is a primitive root
rec(
primes := [ 3 ],
exponents := [ 2 ],
generators := [ 2 ] )
gap> GeneratorsPrimeResidues( 24 );     # 8 divides 24
rec(
primes := [ 2, 3 ],
exponents := [ 3, 1 ],
generators := [ [ 7, 13 ], 17 ] )
gap> GeneratorsPrimeResidues( 1155 );
rec(
primes := [ 3, 5, 7, 11 ],
exponents := [ 1, 1, 1, 1 ],
generators := [ 386, 232, 661, 211 ] )

15.8 GaloisGroup for Number Fields

The Galois automorphisms of the cyclotomic field Qn are given by linear extension of the maps * k: en → enk with 1 ≤ k < n and Gcd( n, k ) = 1 (see GaloisCyc). Note that this action is not equal to exponentiation of cyclotomics, i.e., in general z* k is different from zk:

    gap> ( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 );
-2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
E(5)^2+E(5)^3

For Gcd( n, k ) ≠ 1, the map en → enk is not a field automorphism but only a linear map:

    gap> GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 );
2
-6

The Galois group Gal( Qn, Q ) of the field extension Qn/Q is isomorphic to the group (Z/nZ)* of prime residues modulo n, via the isomorphism

 (Z/nZ)* → Gal( Qn, Q ) k → ( z → z* k )
,

thus the Galois group of the field extension Qn / L with L ⊆ Qn which is simply the factor group of Gal( Qn, Q ) modulo the stabilizer of L, and the Galois group of L/L which is the subgroup in this group that stabilizes L, are easily described in terms of (Z/nZ)* (Generators of (Z/nZ)* can be computed using GeneratorsPrimeResidues GeneratorsPrimeResidues.).

The Galois group of a field extension can be computed using GaloisGroup GaloisGroup:

    gap> f:= NF( [ EY(48) ] );
NF(48,[ 1, 47 ])
gap> g:= GaloisGroup( f );
Group( NFAutomorphism( NF(48,[ 1, 47 ]) , 17 ), NFAutomorphism( NF(48,
[ 1, 47 ]) , 11 ), NFAutomorphism( NF(48,[ 1, 47 ]) , 17 ) )
gap> Size( g ); IsCyclic( g ); IsAbelian( g );
8
false
true
gap> f.base; g.1; f.base ^ g.1;
E(24)-E(24)^11
NFAutomorphism( NF(48,[ 1, 47 ]) , 17 )
E(24)^17-E(24)^19
gap> Operation( g, NormalBaseNumberField( f ), OnPoints );
Group( (1,6)(2,4)(3,8)(5,7), (1,4,8,5)(2,3,7,6), (1,6)(2,4)(3,8)
(5,7) )

The number field automorphism NFAutomorphism( F, k ) maps each element x of F to GaloisCyc( x, k ), see GaloisCyc.

15.9 ZumbroichBase

ZumbroichBase( n, m )

returns the set of exponents i where eni belongs to the base {B}n,m of the field extension Qn/Qm; for that, n and m must be positive integers where m divides n.

{B}n,m is defined as follows:

Let P denote the set of prime divisors of n, n = ∏p∈ P pνp, m = ∏p∈ P pμp with μp ≤ νp, and { en1j}j∈ J ⊗ { en2k}k∈ K = { en1j . en2k}j∈ J, k∈ K.

Then

{B}n,m = p∈ P k=μpνp-1 { epk+1j}j∈ Jk,p \rm where Jk,p = {
 { 0 } ; k=0, p=2 { 0, 1 } ; k > 0, p=2 { 1, ..., p-1 } ; k = 0, p≠ 2 { -(p-1)/(2), ..., (p-1)/(2) } ; k > 0, p≠ 2
. .

{B}n,1 is equal to the base {B}(Qn) of Qn over the rationals given in Zum89 (Note that the notation here is slightly different from that there.).

{B}n,m consists of roots of unity, it is an integral base (that is, the integral elements in Qn have integral coefficients, see Cyclotomic Integers), it is a normal base for squarefree n and closed under complex conjugation for odd n.

    gap> ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );
[ 1, 2, 4, 7, 8, 11, 13, 14 ]
[ 0, 3 ]
gap> ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );
[ 2, 4, 6, 8 ]
[ 0, 1, 2, 3, 4, 5, 6, 7 ]

15.10 Integral Bases for Number Fields

LenstraBase( n, stabilizer, super )

returns a list [  b1, b2, ..., bm  ] of lists, each bi consisting of integers such that the elements j∈ bi E(n)j form an integral base of the number field NF( n, stabilizer ), see Number Field Records.

super is a list representing a supergroup of the group described by the list stabilizer; the base is chosen such that the group of super acts on it, as far as this is possible.

Note: The bi are in general not sets, since for stabilizer = super, bi is always an element of ZumbroichBase( N, 1 ); this is used by NF (see Number Field Records) and Coefficients (see Coefficients for Number Fields).

stabilizer must not contain the stabilizer of a proper cyclotomic subfield of Qn.

    gap> LenstraBase( 24, [ 1, 19 ], [ 1, 19 ] );          # a base of
[ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ]               # Q3(√6),
gap> LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ] );   # another one
[ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ]
gap> LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ) ); # normal base of
[ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]          # Q3(√5)

15.11 NormalBaseNumberField

NormalBaseNumberField( F )
NormalBaseNumberField( F, x )

returns a list of cyclotomics which form a normal base of the number field F (see Number Field Records), i.e. a vector space base of the field F over its subfield F.field which is closed under the action of the Galois group F.galoisGroup of the field extension.

The normal base is computed as described in Art68: Let Φ denote the polynomial of a field extension L/L, Φ its derivative and α one of its roots; then for all except finitely many elements z∈ L, the conjugates of (Φ(z))/((z-α).Φ(α)) form a normal base of L/L.

When NormalBaseNumberField( F ) is called, z is chosen as integer, starting with 1, NormalBaseNumberField( F, x ) starts with z=x, increasing by one, until a normal base is found.

    gap> NormalBaseNumberField( CF( 5 ) );
[ -E(5), -E(5)^2, -E(5)^3, -E(5)^4 ]
gap> NormalBaseNumberField( CF( 8 ) );
[ 1/4-2*E(8)-E(8)^2-1/2*E(8)^3, 1/4-1/2*E(8)+E(8)^2-2*E(8)^3,
1/4+2*E(8)-E(8)^2+1/2*E(8)^3, 1/4+1/2*E(8)+E(8)^2+2*E(8)^3 ]

15.12 Coefficients for Number Fields

Coefficients( z )
Coefficients( F, z )

return the coefficient vector cfs of z with respect to a particular base B, i.e., we have z = cfs * B. If z is the only argument, B is the default base of the default field of z (see DefaultField and Field for Cyclotomics), otherwise F must be a number field containing z, and we have B = F.base.

The default base of a number field is defined as follows:

For the field extension Qn/Qm (i.e. both F and F.field are cyclotomic fields), B is the base {B}n,m described in ZumbroichBase. This is an integral base which is closely related to the internal representation of cyclotomics, thus the coefficients are easy to compute, using only the zumbroichbase fields of F and F.field.

For the field extension L/Q where L is not a cyclotomic field, B is the integral base described in Integral Bases for Number Fields that consists of orbitsums on roots of unity. The computation of coefficients requires the field F.coeffslist.

in future: replace Q by Qm

In all other cases, B = NormalBaseNumberField( F ). Here, the coefficients of z with respect to B are computed using F.coeffslist and F.coeffsmat.

If F.base is not the default base of F, the coefficients with respect to the default base are multiplied with F.basechangemat. The only possibility where it is allowed to prescribe a base is when the field is constructed (see Number Field Records, Cyclotomic Field Records).

    gap> F:= NF( [ ER(3), EB(7) ] ) / NF( [ ER(3) ] );
NF(84,[ 1, 11, 23, 25, 37, 71 ])/NF(12,[ 1, 11 ])
gap> Coefficients( F, ER(3) ); Coefficients( F, EB(7) );
[ -E(12)^7+E(12)^11, -E(12)^7+E(12)^11 ]
[ 11*E(12)^4+7*E(12)^7+11*E(12)^8-7*E(12)^11,
-10*E(12)^4-7*E(12)^7-10*E(12)^8+7*E(12)^11 ]
gap> G:= CF( 8 ); H:= CF( 0, NormalBaseNumberField( G ) );
CF(8)
CF( 0,[ 1/4-2*E(8)-E(8)^2-1/2*E(8)^3, 1/4-1/2*E(8)+E(8)^2-2*E(8)^3,
1/4+2*E(8)-E(8)^2+1/2*E(8)^3, 1/4+1/2*E(8)+E(8)^2+2*E(8)^3 ])
gap> Coefficients( G, ER(2) ); Coefficients( H, ER(2) );
[ 0, 1, 0, -1 ]
[ -1/3, 1/3, 1/3, -1/3 ]

15.13 Domain Functions for Number Fields

The following functions of FieldOps (see chapter Fields) are overlaid in NumberFieldOps:

/, Coefficients, Conjugates, GaloisGroup, in, Intersection, Norm, Order, Print, Random, Trace.

The following functions of NumberFieldOps are overlaid in CyclotomicFieldOps:

Coefficients, Conjugates, in, Norm, Print, Trace.

gap3-jm
11 Mar 2019