Is there another proof for Dirichlet's theorem? 
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Is a “non-analytic” proof of Dirichlet’s theorem on primes known or possible? 

Dirichlet's theorem on primes in arithmetic progression states that there are infinitely many primes of the form $kn+h$ given that $k$ and $h$ are coprime. Is there a short proof for this?
 A: Well, there are short proofs of particular instances of the result. For example, emulating the Euclidean assault on the infinitude of primes, one can establish, almost effortlessly, that there are infinitely many primes of the form 4k+3. Nevertheless, you have to be warned that there is no way to strenghten this technique in order to get the result for every arithmetic progression. You may want to take a look at [1]. In that note, Professor Murty mentions that it was I. Schur the one who first derived a sufficient condition for the existence of an "Euclidean" proof for the infinitude of primes in the arithmetic progression {$mk+a$}$_{k \in \mathbb{N}}$.
Edit: As David Speyer mentioned above one on the main ingredients in the proof is a certain non-vanishing result for L-series. Hence, a way in which one might shorten the proof is by spotting the shortest demonstration for the corresponding non-vanishing theorem. I higly recommend that you take a look at the thread in [2] if you wish to learn more about this particular matter.
References
[1] M. R. Murty, Primes in certain arithmetic progressions, J. Madras. Univ. (1988), 161-169.
[2] Shortest/Most elegant proof for the non-vanishing of $L(1, \chi)$ : 
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
