group theory - Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra?

I would be particularly interested in results from group theory or Galois theory.

In an ideal answer,

1. the theorem should be purely algebraic in nature,




2. the proof should be complex analytic at a crucial step, and




3. the result should be unprovable (or, at least, unproven) by purely algebraic methods.

To elaborate on these criteria, by (1) I mean that the statement of the theorem should be independent of analysis, i.e. not something like "Let $\alpha=$ (some complex integral). Then $G(\mathbb{Q}(\alpha)/\mathbb{Q})=\ldots$". By (2) I mean that I am looking for something where complex numbers are not merely present, but must be used analytically. So, Maschke's theorem for example would not apply just because it involves vector spaces over $\mathbb{C}$. Lastly, (c) primarily means that I am not looking for alternative proofs of known results, no matter how much simpler they may be (no FTA).

Thank you for reading.