 Local realization as a completion of a number field

Every finite extension of ${\mathbb{Q}}_p$ is isomorphic to the completion $F_{\mathfrak{p}}$ of some number field $F$ at a prime $\mathfrak{p}$ above $p$. Approximate a primitive element's minimal polynomial by a polynomial over $\mathbb{Q}$ and use Krasner's lemma to preserve the generated local field.