Here's an exercise I found:
1) Show that $$f(X):=X^3+X+1$$ $$g(X):=X^3+X^2+1 $$ are irreducible in $\mathbb{F_2}[X]$.
2) Let $a, b \in \overline{\mathbb{F_2}}$ with $f(a)=g(b)=0$. $$\mathbb{F_2}\subseteq\mathbb{F_2}(a),\mathbb{F_2}(b)$$ are field extensions. Show that $\mathbb{F_2}(a)$ and $\mathbb{F_2}(b)$ are isomorphic $\mathbb{F_2}$-algebras.
1) is pretty easy to show: Since $\mathbb{F_2}$ is a field, a polynomial of degree $3$ can only be divided into a polynomial of degree $2$ and another one of degree $1$. So it's enough to show that $f(X)$ and $g(X)$ have no roots in $\mathbb{F_2}$. And so $f(1),g(1)=1$ and $f(0),g(0)=1$. Is there anything more to show?
2) I think we only know one theorem about this topic. It says:
If $\mathbb{F_2}(a)$ and $\mathbb{F_2}(b)$ are algebraic closures of $\mathbb{F_2}$, then an isomorphism between $\mathbb{F_2}$(a) and $\mathbb{F_2}(b)$ exists.
This would easily proof 2), but I don't know how to show that these are algebraic closures of $\mathbb{F_2}$.
It'd be great, if anyone could help me.