Link between Riemann surfaces and Galois theory

Solution 1:

The following three categories are equivalent:

1. Smooth projective algebraic curves over $\mathbb{C}$ and nonconstant algebraic maps.
2. Compact connected Riemann surfaces and nonconstant holomorphic maps.
3. The opposite of the category of finitely generated field extensions of $\mathbb{C}$ of transcendence degree $1$ and morphisms of extensions of $\mathbb{C}$.

Some of the functors between these are easy to describe. $1 \Rightarrow 2$ is given by taking the underlying complex manifold, and $2 \Rightarrow 3$ is given by taking the field of meromorphic functions. But the proofs that these are equivalences is nontrivial (I think $1 \Leftrightarrow 2$ is hard and I don't remember how hard $2 \Leftrightarrow 3$ is).

In particular, studying finite extensions of $\mathbb{C}(t)$ is equivalent to studying branched covers of $\mathbb{CP}^1$ (in either the algebraic or the holomorphic categories), which is how Galois theory fits into all of this.