Define the terms function element and complete analytic function.

Let $(f, D)$ be a function element such that $f(z)^{n}=p(z)$, for some integer $n \geqslant 2$, where $p(z)$ is a complex polynomial with no multiple roots. Let $F$ be the complete analytic function containing $(f, D)$. Show that every function element $(\tilde{f}, \tilde{D})$ in $F$ satisfies $\tilde{f}(z)^{n}=p(z) .$

Describe how the non-singular complex algebraic curve

$$C=\left\{(z, w) \in \mathbb{C}^{2} \mid w^{n}-p(z)=0\right\}$$

can be made into a Riemann surface such that the first and second projections $\mathbb{C}^{2} \rightarrow \mathbb{C}$ define, by restriction, holomorphic maps $f_{1}, f_{2}: C \rightarrow \mathbb{C}$.

Explain briefly the relation between $C$ and the Riemann surface $S(F)$ for the complete analytic function $F$ given earlier.

[You do not need to prove the Inverse Function Theorem, provided that you state it accurately.]