I want to find an explicit isomorphism $\mathbb F _{2^4} \cong \mathbb F_{4^2}$, where $\mathbb{F}_{2^4}$ is a quartic extension of $\mathbb{F}_2$, and $\mathbb F_{4^2}$ is a quadratic extension of $\mathbb{F}_4$.
For sure it is $\mathbb F _{2^4} \cong \mathbb F_{4^2}$ by using the classification theorem of finitely generated Abelian groups.
My aim is, to construct $\mathbb F_2 [X] / (f)$ and $\mathbb F_4 [X] / (g)$ with irreducible polynomials $f \in \mathbb F_2 [X]$ and $g \in \mathbb F_4 [X]$ with degree 4, resp. 2, and to give a concrete ismorphism.
Finding $f$ is easy. Just factor $x^{16}-x \bmod 2$ and take any factor of degree $4$: $$ x^{16}-x = x (x + 1) (x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1) (x^4 + x^3 + x^2 + x + 1) $$