Define the concept of separability and normality for algebraic field extensions. Suppose $K=k(\alpha)$ is a simple algebraic extension of $k$, and that $\operatorname{Aut}(K / k)$ denotes the group of $k$-automorphisms of $K$. Prove that $|\operatorname{Aut}(K / k)| \leqslant[K: k]$, with equality if and only if $K / k$ is normal and separable.

[You may assume that the splitting field of a separable polynomial $f \in k[X]$ is normal and separable over $k$.]

Suppose now that $G$ is a finite group of automorphisms of a field $F$, and $E=F^{G}$ is the fixed subfield. Prove:

(i) $F / E$ is separable.

(ii) $G=\operatorname{Aut}(F / E)$ and $[F: E]=|G|$.

(iii) $F / E$ is normal.

[The Primitive Element Theorem for finite separable extensions may be used without proof.]