I want to determine whether the extension $F$ over $\mathbb{F}_2(x)$ is separable, where: $$F=\mathbb{F}_2(x)[t]/ \langle t^2+t+1 \rangle \quad \mbox{and} \quad \mathbb{F}_2 \mbox{ is the finite group with } 2 \mbox{ elements } \cong \mathbb{Z}_2$$
So far, I've worked out that $E=\mathbb{F}_2[x]/ \langle x^2+x+1 \rangle$ is separable over $\mathbb{F}_2$ by checking that for every $\alpha \in E$, the minimal polynomial of $\alpha$ in $\mathbb{F}_2$ is separable. I was able to do this because $|E| = 2^2=4$, but $|F| = \infty$ and therefore I can't use this method anymore. Can someone help me?
This is not how I would do it, but it seems to be the sort of argument you are asking for.
A general element $\alpha$ of your field $F$ is $f(x)\bar{t}+g(x)$. If $f=0$ this element lies in $\mathbb{F}_2(x)$ and so is separable. If $f\not=0$ then the minimal polynomial $m(Y)$ of $\alpha$ is $$ Y^2+f(x)Y+f(x)^2+f(x)g(x)+g(x)^2 $$ whose derivative wrt $Y$ is (the invertible) $f(x)$, so that the highest common factor of $m$ and $m'$ is $1$. Hence $m$ does not have repeated roots and so $\alpha$ is separable.