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There are many proofs that there is a unique division quaternion algebra over a locally compact field that is not $\mathbb{C}$. For instance this set of notes/book by John Voight contains two proofs:

That said, every proof that I've seen first splits the proof into the archimedean and non-archimedean cases (and then sometimes into the residue characteristic 2 and $\ne$ 2 cases).

Even Weil's Basic Number Theory doesn't seem to have a proof that treats all places equally (maybe I missed it?). Does anyone know of such a proof?

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The two cases are different in that the Brauer fields are pretty different. Don't you find treating them separately natural? – Mariano Suárez-Alvarez Mar 10 '11 at 2:17
I find it natural enough for the purposes of proving the fact, but conceptually I'm not entirely sated. Sure the Brauer Groups are quite different, but the subgroup describing the quaternion algebras is the same. Personally I can't help but wondering if this is a fluke or if there's something about local compactness which forces this. – Curious in the USA Mar 10 '11 at 5:26
But you're already excluding a case in order to get a true theorem: namely, the locally compact field $\mathbb{C}$ does not admit a division quaternion algebra. To me, that already suggests that the type of unification you're looking for is unlikely. – Pete L. Clark Mar 10 '11 at 5:37

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