In order to obtain the isomorphic matrix (that map from field $\displaystyle GF(2^8)$ to $\displaystyle GF(((2^2)^2)^2))$, we need a root alpha such that $\displaystyle R(\alpha) = 0$ whereby$\displaystyle R(z) = z^8 +z^4 + z^3 + z + 1$ is a field polynomial of $\displaystyle GF(2^8) $

The representation of $\displaystyle GF(((2^2)^2)^2))$ is generated by

$\displaystyle x^2 + x + {1000}_2$, $\displaystyle x^2 + x + {10}_2$ and $\displaystyle x^2 + x + 1$

so I understand that $\displaystyle \alpha$ range from 0 to 255 and how do I actually test that $\displaystyle R(\alpha) = 0$. how do I actually perform such arithmetic when the element are bind to composite field operations.