I'm trying to grasp proofs of the fact that compact Riemann surfaces are algebraic curves. It seems as though no textbook fully treats this, and I wanted to ask about what was going on. On one hand, I know that one can use Riemann-Roch to furnish meromorphic functions and then an embedding into projective space, whence it is possible to use something like Chow's Theorem to show that the image is algebraic. I would like to ask about a different approach:
I also know that once one shows that nonconstant meromorphic functions exist on any compact Riemann surface (e.g. using Riemann-Roch), it is possible to realize such a meromorphic function as providing a branched cover of $\mathbb{P}^1$; how exactly is this done, and how does this prove algebraicity?
Among others, Hartshorne and Miranda mention that showing existence of nonconstant meromorphic functions is a task in hard analysis: is that in order to prove Riemann-Roch, or is it independent?
Thank you - I would appreciate clarification on this equivalence of categories, and the direction of the implications in question/where certain tools get used.