(a) Define what it means for a finite field extension $L$ of a field $K$ to be separable. Show that $L$ is of the form $K(\alpha)$ for some $\alpha \in L$.

(b) Let $p$ and $q$ be distinct prime numbers. Let $L=\mathbb{Q}(\sqrt{p}, \sqrt{-q})$. Express $L$ in the form $\mathbb{Q}(\alpha)$ and find the minimal polynomial of $\alpha$ over $\mathbb{Q}$.

(c) Give an example of a field extension $K \leqslant L$ of finite degree, where $L$ is not of the form $K(\alpha)$. Justify your answer.