Krasner's lemma

Let $K$ be complete, let $\alpha$ be separable over $K$, and let $\beta$ be algebraic over $K$. If $\beta$ is closer to $\alpha$ than any other $K$-conjugate of $\alpha$, then

The roots of a separable polynomial over a complete non-Archimedean field vary continuously with its coefficients after the roots are suitably matched. In particular, a sufficiently close coefficientwise approximation has one root in each sufficiently small disjoint ball around an original root.

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.