I am trying to create an isomorphism between GF($2^8$) and an extension field in GF($(2^4)^2$). The finite field GF($2^8$) uses the Rijndael(AES) irreducible polynomial.In this field I have found 128 generators(G1).
I have found 3 irreducible polynomials(IR1) in GF($2^4$). For each irreducible polynomial, I have found 120 irreducible polynomials(IR2) in the extension field GF($(2^4)^2$). For each of the 360 pairs of (IR1,IR2) I have found the 128 generators(G2).
So, now I am trying to create an isomorphism using (G1,G2) by mapping $G1^i \to G2^i$ ( by doing multiplications in their respective fields). What I find is that many pairs of (G1,G2) are not forming isomorphisms . This was verified by checking if map(1) +map(2) =map(3).
Is this supposed to happen? Why/Why not? Please explain in detail.
Note:My knowledge in isomorphisms is not great and was unable to find good online sources for the same.