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.
GAP 3.4.4