(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a polynomial of degree $d>0$, and let $m_{1}, \ldots, m_{k}$ be the multiplicities of the ramification points of $f$. Prove that

$$\sum_{i=1}^{k}\left(m_{i}-1\right)=d-1$$

Show that, for any list of integers $m_{1}, \ldots, m_{k} \geqslant 2$ satisfying $(*)$, there is a polynomial $f$ of degree $d$ such that the $m_{i}$ are the multiplicities of the ramification points of $f$.

(b) Let $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ be an analytic map, and let $B$ be the set of branch points. Prove that the restriction $f: \mathbb{C}_{\infty} \backslash f^{-1}(B) \rightarrow \mathbb{C}_{\infty} \backslash B$ is a regular covering map. Given $z_{0} \notin B$, explain how a closed loop $\gamma$ in $\mathbb{C}_{\infty} \backslash B$ gives rise to a permutation $\sigma_{\gamma}$ of $f^{-1}\left(z_{0}\right)$. Show that the group of all such permutations is transitive, and that the permutation $\sigma_{\gamma}$ only depends on $\gamma$ up to homotopy.

(c) Prove that there is no meromorphic function $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ of degree 4 with branch points $B=\{0,1, \infty\}$ such that every preimage of 0 and 1 has ramification index 2 , while some preimage of $\infty$ has ramification index equal to 3. [Hint: You may use the fact that every non-trivial product of $(2,2)$-cycles in the symmetric group $S_{4}$ is a $(2,2)$-cycle.]