In my study of fields, the notion of the separability of an algebraic field extension is one of the more slippery concepts I have encountered thusfar. What is particularly vexing to me is the notion of a separable closure. Let $E/F$ be an algebraic field extension. The separable closure of $F$ in $E$ is defined to be the subfield
$E_{sep}=\{\alpha \in E \; | \; \alpha $ is separable over $ F \}$.
While the concept of a separable closure is easy enough to understand based on the definition, the actual calculation of a separable closure $E_{sep}$ for a given algebraic field extension $E/F$ is not obvious to me.
For example, I was recently assigned the following problem:
Let $k=\mathbb{F}_2(t)$, $\alpha$ be a root of $X^4+tX^2+t\in k[X]$ in an algebraic closure of $k$ and $K=k(\alpha)$, where $\mathbb{F_2}=\mathbb{Z}/2\mathbb{Z}$. Determine $K_{sep}$ and $[K_{sep}:K]$.
My first thought is that since the separability of an element $a$ of the algebraic field extension $E/F$ is defined based on the structure of its minimal polynomial $m_{a,F}(X) \in F[X]$, we can use this to determine the elements of $E_{sep}$. An element $e \in E$ is inseparable if and only if $m_{e,F}(X)=g(X^p)$ for some separable $g(X) \in F[X]$. So the separable elements of $E$ are those whose minimal polynomial is not an element of $F[X^p]$. This, however, does not seems to be a tractable idea.
My questions are
How does one approach the problem of calculation the separable closure of an algebraic extension, like, for example, the one above?
If there is no general, one-size-fits-all method, what are some examples of techniques used to deduce the separable closure?
Note: any questions asked herein are not current homework questions.