Equivalent Definition of Non-Archimedean Local Field 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 $L$?

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.
