Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and $F=\mathbb{Z}_2(\beta)$. I am asked to find an explicit isomorphism $\phi: K \rightarrow F$.
I know that since $p(\alpha)=0$ and $q(\beta)=0$ then $\alpha$ and $\beta$ are algebraic over $\mathbb{Z}_2$. I'm rather stuck after this point, and my other problems are somewhat similar to this one, so if there is a key fact that I'm missing then it might allow me to solve the others.