= Solution
A <separable algebraic element> has a <minimal polynomial> with distinct roots in an <algebraic closure>. A <separable field extension> is an algebraic extension all of whose elements are separable. In characteristic $p$, the extension $\mathbb F_p(t)/\mathbb F_p(t^p)$ is inseparable: the minimal polynomial of $t$ is $T^p-t^p$, irreducible over the smaller field but having the single root $t$ with multiplicity $p$.
If $L$ is a <finite field> of order $q$, every element of $L$ is a root of $T^q-T\in K[T]$, because $K$ contains its prime field. This polynomial has derivative $-1$, so it has no repeated roots. Every element's minimal polynomial divides it and is therefore separable. Hence \b[$L/K$ is separable].
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