---
tag: G8PGN0WQHW
kind: statement
role: lemma
name: Krasner's lemma
msc: [12J25, 11S15]
uses: [1H5F1BP6XS, KFYW0WN9AD, 0MCXX29HWY, 0N019F6RE9]
about: [1H5F1BP6XS, 3ACEE4N6NE]
sources:
  - url: https://en.wikipedia.org/wiki/Krasner%27s_lemma
same_as:
  wikidata: Q6436234
  wikipedia: Krasner's lemma
---

## Statement

Let $K$ be a field that is complete for a [[non-Archimedean absolute value|1H5F1BP6XS]] $\abs{\cdot}$.
Let $\overline{K}$ be an [[algebraic closure|0MCXX29HWY]] of $K$, with the unique extension of $\abs{\cdot}$ to $\overline{K}$.
Let $\alpha \in \overline{K}$ be [[separable|0N019F6RE9]] over $K$, and let $\alpha = \alpha_1, \alpha_2, \ldots, \alpha_n$ be the distinct conjugates of $\alpha$ over $K$, with $n \ge 2$.
Let $\beta \in \overline{K}$ satisfy
$$\abs{\beta - \alpha} < \abs{\alpha - \alpha_i} \quad \text{for } i = 2, \ldots, n.$$
Then $K(\alpha) \subseteq K(\beta)$.

## 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|FKF9YGBKHF]] form an algebraically closed field.

## Explanations

### L1 human

In a $p$-adic field, let $\beta$ be closer to $\alpha$ than any other conjugate of $\alpha$ is.
Then $\beta$ already "knows" $\alpha$: the field made from $\beta$ contains $\alpha$.
Completeness is needed.
In $\Q$ with the $7$-adic absolute value, rational numbers can be as close as one likes to a square root of $2$.
Yet they generate only $\Q$, which does not contain that square root.

### L2 human

Krasner's lemma is the basic continuity statement for roots in local fields.
With the [[Newton polygon|H7QNCE2QZJ]] and [[Hensel's lemma|010QVB71T2]] 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.