Let $L$ be a finite field extension of a field $K$, and let $G$ be a finite group of $K$ automorphisms of $L$. Denote by $L^{G}$ the field of elements of $L$ fixed by the action of $G$.

(a) Prove that the degree of $L$ over $L^{G}$ is equal to the order of the group $G$.

(b) For any $\alpha \in L$ write $f(t, \alpha)=\Pi_{g \in G}(t-g(\alpha))$.

(i) Suppose that $L=K(\alpha)$. Prove that the coefficients of $f(t, \alpha)$ generate $L^{G}$ over $K$.

(ii) Suppose that $L=K\left(\alpha_{1}, \alpha_{2}\right)$. Prove that the coefficients of $f\left(t, \alpha_{1}\right)$ and $f\left(t, \alpha_{2}\right)$ lie in $L^{G}$. By considering the case $L=K\left(a_{1}^{1 / 2}, a_{2}^{1 / 2}\right)$ with $a_{1}$ and $a_{2}$ in $K$, or otherwise, show that they need not generate $L^{G}$ over $K$.