Is there a nontrivial field homomorphism from $GF(2^n)$ to $ GF(2)^n$?

Question

Does there exist a non trivial field homomorphism $\phi: GF(2^n) \rightarrow GF(2)^n$?

Where $GF(2)^n$ is $GF(2) \times GF(2)$ repeated n times.

Answer 1

Assuming you ask for ring homomorphisms, the answer is that no such homomorphism exists. Indeed, any such homomorphism corresponds to $n$ homomorphisms $\phi_i\colon GF(2^n)\to GF(2)$. For $n\geq 2$, such homomorphisms do not exist since if there were any such homomorphism it would be injective, but clearly no injective map between both fields can exist.