Let $F$ be a field of characteristic $2$. Find the maximal separable subextension in $F(X)/F(X^4 + X^2)$.
I am not sure what to do here. I know that if $f(X) = aX^3 + bX^2 + cX + d \in \mathbb{F}_2[X]$, then $f'(X) = aX^2 + c$, so that if $f(X)$ were a separable extension, then at least one of $a$ or $c$ has to be nonzero. However, I'm not sure what to do with polynomials in $X$ of higher degrees, much less solving the case for this $f$.
The entire extension has degree $4$, since $X$ is a root of $Y^4 + Y^2 - (X^4 + X^2)$. Furthermore, we can factor $Y^4 + Y^2 - (X^4 + X^2)$ as $(Y^2 + Y - (X^2+ X))^2$, and so the extension is inseparable. Therefore we claim that $F(X^2)/F(X^4 + X^2)$ is the maximum separable subextension. Observe that if this extension is separable and nontrivial, then it is maximal since the entire extension is inseparable. Indeed it is separable and nontrivial, since it is a root of $Y^2 - Y - (X^4 + X^2)$, and it has degree $2$ since this polynomial is irreducible (you can show this by thinking about quadratic extensions in characteristic $2$). The proof follows.