Krasner's lemma

Remarks

Proof idea: let $\sigma$ be an automorphism of $\overline{K}$ that fixes $K(\beta)$. By uniqueness of the extended absolute value, $\abs{\sigma(x)} = \abs{x}$ for all $x$, so $\abs{\sigma(\alpha) - \beta} = \abs{\sigma(\alpha - \beta)} = \abs{\alpha - \beta}$. The strong triangle inequality gives $\abs{\sigma(\alpha) - \alpha} \le \max(\abs{\sigma(\alpha) - \beta}, \abs{\beta - \alpha}) < \abs{\alpha - \alpha_i}$ for $i \ge 2$, so $\sigma(\alpha) = \alpha$. As $\alpha$ is separable and fixed by all such $\sigma$, it lies in $K(\beta)$. The same proof works for a Henselian valued field, since there too the absolute value extends uniquely. For $n = 1$ the element $\alpha$ lies in $K$, so the conclusion holds for every $\beta$.

Application: let $f$ be a monic irreducible separable polynomial over $K$, and let $g$ be a monic polynomial over $K$ of the same degree with close enough coefficients. Then $g$ is irreducible, and each root of $g$ generates the same extension of $K$ as a suitable root of $f$. This shows that every finite extension of $\Q_p$ is generated by a root of a polynomial with rational coefficients. It is also the key step in the proof that the complex p-adic numbers form an algebraically closed field.

Going deeper

For a student

Krasner's lemma is the basic continuity statement for roots in local fields. With the Newton polygon and Hensel's lemma it is one of the three main tools for studying extensions of $\Q_p$. It implies that a finite separable extension of $\Q_p$ does not change when the defining polynomial is perturbed slightly. With a compactness argument, this shows that $\Q_p$ has only finitely many extensions of each degree. It also gives that the completion of an algebraic closure of $\Q_p$ is again algebraically closed.