Local realization as a completion of a number field
= 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.