Define the degree of a non-constant holomorphic map between compact connected Riemann surfaces and state the Riemann-Hurwitz formula.

Show that there exists a compact connected Riemann surface of any genus $g \geqslant 0$.

[You may use without proof any foundational results about holomorphic maps and complex algebraic curves from the course, provided that these are accurately stated. You may also assume that if $h(s)$ is a non-constant complex polynomial without repeated roots then the algebraic curve $C=\left\{(s, t) \in \mathbb{C}^{2}: t^{2}-h(s)=0\right\}$ is path connected.]