 Closest-root descent in a Galois extension

Let $L/K$ be a finite Galois extension of a field complete for a Non-Archimedean absolute value. Let $\beta \in L$. Suppose $\alpha \in L$ 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 ${\mathbb{Q}}_p$.