GAP Manual: 6.11. Trace

Trace( z )
Trace( F, z )

In the first form Trace returns the trace of the field element z in its default field over its prime field (see DefaultField). In the second form Trace returns the trace of the element z in the field F over the subfield F.field.

The trace of an element z in a field F over a subfield S is the negative of the coefficient of the second highest degree term of the characteristic polynomial of z (see CharPol). Thus the trace lies in S. The trace is the sum over all conjugates of z in the normal closure of F over S (see Conjugates).

    gap> Trace( Z(2^6) );
    0*Z(2)
    gap> Trace( GF(2^12), Z(2^6) );
    0*Z(2)
    gap> Trace( GF(2^12)/GF(2^2), Z(2^6) );
    0*Z(2) 

The default function FieldOps.Trace adds the conjugates of z in F (see Conjugates). For nonabelian extensions, this is overlayed by a function, which obtains the trace from the characteristic polynomial.