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True. Put $H=\mathrm{Gal}(K/{\mathbb{Q}}_p(\beta ))$. Every $\sigma \in H$ fixes $\beta$, permutes the polynomial's polynomial roots, and is an isometry by the uniqueness proved in part (a). ThereforeThe strict inequalities single out ${\alpha }_1$ as the unique closest polynomial root, so $\sigma ({\alpha }_1)={\alpha }_1$ for every $\sigma \in H$. The Fundamental theorem of Galois theory then givesThis closest-root descent in a Galois extension is the relevant form of Krasner's lemma. If polynomial roots repeat, the strict hypothesis excludes a second occurrence of ${\alpha }_1$; otherwise the same argument applies unchanged.