Let $L$ be a field, and $G$ a group which acts on $L$ by field automorphisms.

(a) Explain the meaning of the phrase in italics in the previous sentence.

Show that the set $L^{G}$ of fixed points is a subfield of $L$.

(b) Suppose that $G$ is finite, and set $K=L^{G}$. Let $\alpha \in L$. Show that $\alpha$ is algebraic and separable over $K$, and that the degree of $\alpha$ over $K$ divides the order of $G$.

Assume that $\alpha$ is a primitive element for the extension $L / K$, and that $G$ is a subgroup of $\operatorname{Aut}(L)$. What is the degree of $\alpha$ over $K$ ? Justify your answer.

(c) Let $L=\mathbb{C}(z)$, and let $\zeta_{n}$ be a primitive $n$th root of unity in $\mathbb{C}$ for some integer $n>1$. Show that the $\mathbb{C}$-automorphisms $\sigma, \tau$ of $L$ defined by

$$\sigma(z)=\zeta_{n} z, \quad \tau(z)=1 / z$$

generate a group $G$ isomorphic to the dihedral group of order $2 n$.

Find an element $w \in L$ for which $L^{G}=\mathbb{C}(w)$.