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 Q_n (over the rationals),

CyclotomicField( gens ):

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

CyclotomicField( subfield, n ):

the field Q_n 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 Q_n/Q_m, and containing Zumbroichbase( n, 1 ) if F is an extension of a number field that is not cyclotomic; Coefficients for Number Fields

Number Field Records).

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