Let $L / K$ be an algebraic extension of fields, and $x \in L$. What does it mean to say that $x$ is separable over $K$ ? What does it mean to say that $L / K$ is separable?

Let $K=\mathbb{F}_{p}(t)$ be the field of rational functions over $\mathbb{F}_{p}$.

(i) Show that if $x$ is inseparable over $K$ then $K(x)$ contains a $p$ th root of $t$.

(ii) Show that if $L / K$ is finite there exists $n \geqslant 0$ and $y \in L$ such that $y^{p^{n}}=t$ and $L / K(y)$ is separable.

Show that $Y^{2}+t Y+t$ is an irreducible separable polynomial over the field of rational functions $K=\mathbb{F}_{2}(t)$. Find the degree of the splitting field of $X^{4}+t X^{2}+t$ over $K$.