(a) Define the notions of (abstract) Riemann surface, holomorphic map, and biholomorphic map between Riemann surfaces.

(b) Prove the following theorem on the local form of a holomorphic map.

For a holomorphic map $f: R \rightarrow S$ between Riemann surfaces, which is not constant near a point $r \in R$, there exist neighbourhoods $U$ of $r$ in $R$ and $V$ of $f(r)$ in $S$, together with biholomorphic identifications $\phi: U \rightarrow \Delta, \psi: V \rightarrow \Delta$, such that $(\psi \circ f)(x)=\phi(x)^{n}$, for all $x \in U$.

(c) Prove further that a non-constant holomorphic map between compact, connected Riemann surfaces is surjective.

(d) Deduce from (c) the fundamental theorem of algebra.