= 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>. Suppose $\alpha$ is uniquely closest to $\beta$ among the roots of a polynomial over $K$ lying in $L$. Every element of $\operatorname{Gal}(L/K(\beta))$ 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 $\mathbb Q_p$.
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