GAP Manual: 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.