How do you find the factors of $x^4+x+1$ in $GF(2^8)$ in terms of polynomials?
Let me explain, We have primitive irreducible polynomial $p(x)=x^2+x+1$ in $GF(2^2)$ which has root $\alpha^2+\alpha$ in $GF(2^4)$.
Example: Using $\alpha^2+\alpha$ in $p\left(\alpha^2+\alpha\right)=0 \mod x^4+x+1$ (where $x^4+x+1$ is primitive irreducible polynomial in $GF(2^4)$.
Then how we can find $x^4+x+1$ roots in $GF(2^8)$ that generate subgroup of degree $15$ which is simple emending of $GF(2^4)$ in $GF(2^8)$?