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I am looking for the original reference for Ostrowski's theorem of 1916 that the only valuations on the rational numbers are the trivial, Archimedean, and p-adic valuations.'s_theorem

Wikipedia refers to Koblitz (p-adic Numbers, p-adic Analysis, and Zeta-Functions), but I can't find the reference in there.

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The reference to the Acta Math. paper is at the Wikipedia page, but with the wrong year (1918). – Barry Cipra May 23 '13 at 18:45
up vote 4 down vote accepted

Barry Cipra is right. See I've changed the wikipedia's entry.

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Ostrowski, Alexander Über einige Lösungen der Funktionalgleichung ψ(x)⋅ψ(x)=ψ(xy). (German) Acta Math. 41 (1916), no. 1, 271–284.

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According to the book of Helmut Koch "Zahlentheorie" also the following refereence is of importance, because there $p$-adic valuations and completions are studied before. The paper is: Über sogenannte perfekte Körper, in Journal der reinen und angewandten Mathematik $147 (1917), 191-204$.

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