Let $L=\mathbf{C}(z)$ be the function field in one variable, $n>0$ an integer, and $\zeta_{n}=e^{2 \pi i / n}$.

Define $\sigma, \tau: L \rightarrow L$ by the formulae

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

and let $G=\langle\sigma, \tau\rangle$ be the group generated by $\sigma$ and $\tau$.

(i) Find $w \in \mathbf{C}(z)$ such that $L^{G}=\mathbf{C}(w)$.

[You must justify your answer, stating clearly any theorems you use.]

(ii) Suppose $n$ is an odd prime. Determine the subgroups of $G$ and the corresponding intermediate subfields $M$, with $\mathbf{C}(w) \subseteq M \subseteq L$.

State which intermediate subfields $M$ are Galois extensions of $\mathbf{C}(w)$, and for these extensions determine the Galois group.