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 Q_n (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!

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