In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved:
Let $U,V \subseteq \mathbb{C}$ be open subsets. Every $\mathbb{C}$-algebra homomorphism $\varphi : K(U) \rightarrow K(V)$ is induced from a holomorphic function $h : V \rightarrow U$: $$\varphi(f) = f \circ h, \; \; f \in K(U).$$ In particular, $U$ and $V$ are biholomorphic if and only if $K(U)$ and $K(V)$ are isomorphic $\mathbb{C}$-algebras.
Here $K(U)$ is the field of meromorphic functions on $U$. There are also generalizations to Stein manifolds.
This looks like a very nice structural result, and the proof is simple enough for an undergraduate level. Annoyingly, I can't think of any interesting consequences of it, in the sense that every $\mathbb{C}$-algebra homomorphism from $K(U)$ to $K(V)$ I can think of is already of this form. I would be grateful for any examples or references you can give me.
