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Wikipedia states that there is an equivalent definition of non-archimedean local fields: "it is a field that is complete with respect to a discrete valuation and whose residue field is finite." However, I'm unable to find any proof or reference for this.

In particular, I'm interested in the following problem: let $K$ be a non-archimedean local field of characteristic 0 (as per the conventional definition) which is a finite extension of $\mathbb Q_p$. How can one prove that $K$ is the completion of $L$ at some place $v$ for some number field $L$?

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Regarding your first question: This is Proposition 1 in Section 1 of Chapter II of Serre's Local Fields.

Regarding the second question: This is a consequence of Krasner's Lemma.

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    $\begingroup$ +1 for mentioning Krasner's lemma - it's left out in far too many elementary texts on algebraic number theory, which is a pity because it's very easy to prove... $\endgroup$ – Daniel Miller Jul 11 '13 at 22:34
  • $\begingroup$ Though I should mention: some authors consider fields complete with respect to a discrete valuation, with a perfect (not necessarily finite) residue field. $\endgroup$ – Daniel Miller Jul 11 '13 at 22:35

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