I just want to check a small thesis of a paper [1]. It is written in the proof of Lemma 4. The author points out, that $\Phi_{12}(p)=p^4-p^2+1$. I wrote a small sageworksheet for verify this, but it gave me something different, so I'm probably wrong. In sage you can calculate through
from sage.rings.polynomial.cyclotomic import cyclotomic_coeffs
c=cyclotomic_coeffs(12)
R = GF(2^(12))['x']
Phi(x)=R(c)
the 12-th cyclotomic polynomial ofer $GF(2^{12})$. Now you can evaluate at p:
var('p');
Phi(p)
and it returns $p^4 + p^2 + 1$. Since the characteristic of R is 2, isn't $p^2 = -p^2$? Or how do I get the statement?