Let $K$ be imperfect, $K^a$ its algebraic closure and $K^{\rm sep}$ its separable closure. Show $[K^a \colon K]$ and $[K^a\colon K^{\rm sep}]$ are infinite. Is $[K^{\rm sep}\colon K]$ infinite?
Since $K$ is not perfect, I know there is an element $a$ in $K$ that has no $p$th root in $K$, i.e. there is an $a$ in $K$ such that there is a $b$ in $K^a\setminus K$ such that $b^p = a$. Also, $x^{p^n} - a$ is irreducible over $K[x]$ for $a$ in $K\setminus K^p$.
I'm pretty sure that I have to assume these degrees are finite and somehow lead to a contradiction from the fact that $K$ is imperfect, but I'm stuck.
Any help would be greatly appreciated.