Traces of finite fields elements lifted to integers

["Emmanuel ROYER, LMBP" <emmanuel.royer@math.cnrs.fr>]

Hello!

I would like to compute traces of finite fields elements.

Let us take F_8 as an example.

The following is obviously false but I am not able to see why. Is not conjvec(z^4) supposed to have 3 elements?

? T=ffinit(2,3)
%1 = Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2)
? t=ffgen(T,'t)
%3 = t
? z=ffprimroot(t)
%4 = t^2
? F8=concat([0],[z^d | d <- [1..7]])
%5 = [0, t^2, t^2 + t + 1, t^2 + t, t, t^2 + 1, t + 1, 1]
? tr(h)=vecsum(conjvec(h))
%6 = (h)->vecsum(conjvec(h))
? apply(tr,F8)
%7 = [0, t + 1, t^2 + 1, t^2 + 1, t^2 + t, t + 1, t^2 + t, 0]
? conjvec(z^4)
%8 = [t, t^2]~

(Indeed the point is that I would like to be able to lift the image of the trace to integers).

Best regards,
Emmanuel