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It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice exposition is

http://math.stanford.edu/~conrad/248APage/handouts/localglobal.pdf

My question is, does anyone know a published reference for this fact?

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    $\begingroup$ The only case that isn't trivial is finite extensions of Q_p, and then it's just immediate from Krasner's Lemma isn't it? Extension is separable so simple, take min poly, adjust so that coefficients are in Q, and done. $\endgroup$ Commented May 20, 2016 at 14:35
  • $\begingroup$ Yes, this is exactly as Lorenz proves the statement in the textbook that I quoted below. $\endgroup$ Commented May 20, 2016 at 16:09
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    $\begingroup$ Might I ask (well, I will): why do you need a published source for this result? $\endgroup$ Commented May 20, 2016 at 17:43
  • $\begingroup$ @KConrad: I am writing a book on automorphic representations with H. Hahn and I want to state this with a reference but not prove it. $\endgroup$ Commented May 21, 2016 at 15:11
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    $\begingroup$ A two line proof as in wrigley's answer would surely be more useful to your readers than a reference. Readers who don't know Krasner's lemma can look it up in the Wikipedia. $\endgroup$ Commented May 21, 2016 at 15:15

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F. Lorenz: Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics, Theorem 2 p. 78.

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