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,


*

*the theorem should be purely algebraic in nature,

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

*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.
 A: Take Dirichlet's theorem on arithmetic progressions, or Chebotarev density; I'm sure there are plenty of other analytic results in number theory that will also work. 
Sure, theorems proven with L-functions are often density statements, which are analytic in nature, but those can be weakened to existence statements; ie. there exist infinitely many primes of the form $a+nk$ when $(a,n)=1$, and there exist infinitely many primes in an abelian extension with a given Frobenius element.
EDIT: I see on wikipedia that there is an elementary proof known for Dirichlet's theorem. But it came along over 100 years after the $L$-funtion proof.
A: To quote Hartshorne from "Algebraic Geometry":

If $X$ is a nonsingular variety over $\mathbf{C}$, then we can also consider $X$ as a complex manifold. All of the methods of complex analysis and differential geometry can be used to study this complex manifold. ... This is an extremely powerful method, which has produced and is still producing many important results, proved by these so-called "transcendental methods," for which no purely algebraic proofs are known.

He gives a example of using the exact sequence
$$ 0 \to \mathbf{Z} \to \mathbf{C} \xrightarrow{f} \mathbf{C}^* \to 0 $$
where $f(x) = e^{2 \pi i x}$ and studying the cohomology of a complex variety by comparing it to the cohomology of the corresponding complex manifold. Alas, I don't really know enough about the subject to be able to say much about it.
A: Sharp bounds on character sums (like Kloosterman sums) often follow from the Riemann hypothesis for suitable $L$-functions of varieties over finite fields, and a piece of the proof of this relies on nonvanishing theorems that ultimately go back to analytic ideas from the proof of the prime number theorem/Dirichlet's theorem.
A: Fermat's Last Theorem. Ultimately, one shows that there are no soutions to a Fermat equation because there are no holomorphic differential forms on $\hat{\mathbb{C}}.$
