It's been ages since I've posted anything of a serious but non-political nature, so I thought I'd spice things up a bit by posting the following proof from a Bulletin of the AMS review of some books on pro-finite groups.

We define theUsing the profinite topology on the ring of integers to prove such a simple theorem is clearly the mathematical equivalent of killing a fly with a cannon - if you know what a profinite group is, you could probably think up several other proofs of the theorem in your sleep - but I still think it's neat how a proof like this demonstrates the underlying unity of the subject.profinite topologyon the ring of integersZby taking as a basis of open sets all the arithmetic progressions. As the complement of an arithmetic progression is itself a union of arithmetic progressions, it is clear that every arithmetic progression is both open and closed. Now, if we assume that there are only finitely many primesp, then the union of the finite number of arithmetic progressionspZis also closed, and hence its complement {+1, -1} is an open set, which is obviously impossible given our chosen basis.

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