I have to prove $p(x)=x^4+x^3+x^2+x+1$ is not primitive polynomial. (Its root is not primitive element of $GF(2^4)$.)
I have shown that it is irreducible. Next my idea is to assume that $p(x)$ is a primitive polynomial.
I think that $n=4$ and $p=2$, hence I need to check $1+(2^4−1)=16$ cases. Should I check, if $p(x)$ has a root in $F_2$, $F_4$ or $F_8$ ? I think, I have to find a degree $k<15$, where $x^k\mod(p(x))=1$ ? What do I have to do next?