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

Let $L/K$ be a finite Galois extension of a field complete for a Non-Archimedean absolute value. Suppose $\alpha$ is uniquely closest to $\beta$ among the roots of a polynomial over $K$ lying in $L$. Every element of $\mathrm{Gal}(L/K(\beta ))$ preserves absolute values on a field and permutes these roots. It sends $\alpha$ to an equally close root, hence fixes $\alpha$. The Fundamental theorem of Galois theory gives $K(\alpha )\subseteq K(\beta )$. The converse inclusion need not hold: for roots $0,\sqrt{p},-\sqrt{p}$, the element $p\sqrt{p}$ is uniquely closest to zero but does not belong to ${\mathbb{Q}}_p$.

A p-adic field has only finitely many extensions of a fixed degree up to isomorphism. There are finitely many possible residue degrees and a unique unramified extension of each such degree. Over each maximal unramified subfield, a totally ramified extension is generated by a root of an Eisenstein polynomial. The coefficient space of these polynomials of fixed degree is compact. Hensel lemma and Krasner's lemma show that sufficiently close polynomials generate isomorphic extensions, giving a finite cover by neighborhoods of a constant extension type.

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.

The completion ${\mathbb{C}}_p$ of an algebraic closure of ${\mathbb{Q}}_p$ is an algebraically closed field. Approximate a polynomial over ${\mathbb{C}}_p$ by one over ${\overline{\mathbb{Q}}}_p$, use continuity of roots over a non-Archimedean field to obtain a nearby algebraic root, and then use Krasner's lemma to show that the original root already belongs to ${\mathbb{C}}_p$.