Since all the finite field of $p^n$ elements are the splitting field of the separable polynomial $x^{p^n}-x$, all of them are isomphic. In particular if $f_1(x),f_2(x)$ are irreducible polynomials over $\mathbb{F}_p[x]$ of the same degree. Then: $ \mathbb{F}_p[x]/(f_1(x)) \cong \mathbb{F}_p[x]/(f_2(x))$ But I want to find an explicit isomorphism. I don't know if it's always possible. But the following could be useful.
Let's consider $f_2$ as a polynomial in $(\mathbb{F}_p[x]/(f_1(x)))[y]$. If $\gamma(x) $ is a root of $f_2(y)$ (a root on $\mathbb{F}_p[x]/(f_1(x))$). Then the following map is an isomorphism: $\mathbb{F}_p[x]/(f_2(x)): \to \mathbb{F}_p[x]/(f_1(x)) $ $x\to \gamma(x)$
My question if there are techniques to find that $\gamma(x)$.
For example if $f_1 = x^4+x^3+1 , f_2 = x^4+x+1 $ are over $\mathbb{F}_2$ then $\gamma(x)=x^3+x^2 $ it's a root.
If the solution of the general case it's not possible (or unsolved) or too difficult, I want to know at least this particular case :/ I want to compute it in the case $ f_1= x^2+2x+2 , f_2 = x^2+x+3 $ over $\mathbb{F}_7$