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

The representation of GF(((2^2)^2)^2)) is generated by

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

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