Let $f(x)=x^2+x+1\in \mathbb{F}_2[x]$ and $\mathbb{F}_4:= \mathbb{F}_2[x]/(f(x))$. Let $\alpha$ be the element $(01)$ in $\mathbb{F}_4$, $\alpha=x$ in polynomial form.
Show that $\mathbb{F}_4[x]/(x^2+x+\alpha)$ is isomorphic to the field $\mathbb{F}_{16}=\mathbb{F}_2[x]/(x^4+x^3+1)$ by explicitly constructing an isomorphism between the fields.