Considering three isomorphism finite fields of size 256 A, B and C; where A and B are $GF(2^8)/GF(2)$ constructed with two distinct irreducible polynomials of degree 8, and C is a tower field $GF(((2^2)^2)^2)$.
Having tow matrices M1 and M2, where M1 (respectively M2) maps the members of A (respectively B) to their corresponding member of B (respectively C); does $M3=M2*M1$ maps the members of A to their corresponding member of C?
Any answer would be appreciated.