(a) Let $K \subseteq L$ be a finite separable field extension. Show that there exist only finitely many intermediate fields $K \subseteq F \subseteq L$.

(b) Define what is meant by a normal extension. Is $\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{1+\sqrt{7}})$ a normal extension? Justify your answer.

(c) Prove Artin's lemma, which states: if $K \subseteq L$ is a field extension, $H$ is a finite subgroup of $\operatorname{Aut}_{K}(L)$, and $F:=L^{H}$ is the fixed field of $H$, then $F \subseteq L$ is a Galois extension with $\operatorname{Gal}(L / F)=H$.