Question 1
Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely inseparable.
In this case, is $E/K$ always separable? If not, what is a counterexample?
Moreover, let $K^s$ and $F^s$ be the separable closures in $E$. Then, $E/K^s$ is purely inseparable. How different are $K^s$ and $F^s$ as fields? That is, how much nicer is the extension $K^s/K$ compared to $F^s/F$?
Question 2
Let $E/F$ be a normal field extension. Let $E^G$ denote the fixed field of $\operatorname{Aut}(E/F)$. Then, $E/E^G$ is separable and $E^G/F$ is purely inseperable.
Is $E^G$ the unique such subextension? That is, is there a subextension $K$ of $F$ such that $E/K$ is separable and $K/F$ is purely inseparable, but $K\neq E^G$?
The extension $E/K$ need not be separable. Here is the example I learned from a note by J. Lipman.
Consider the rational function field $F=\mathbb{F}_2(y,z)$ and the extension $E=F(x)$, where $x$ is a root of $$f(t)=t^4+yt^2+z\in F[t].$$ If $E/K$ was separable, we would have $f=g^2$, for $g\in K[t]$. We have $g=t^2+\sqrt{y}t+\sqrt{z}$, which means that $\sqrt{x},\sqrt{y}\in K$. The latter condition says that $K/F$ is a degree-four extension, and so it is not purely inseparable.