Krasner's lemma

In number theory, more specifically in p-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to its algebraic extensions.

Statement
Let K be a complete non-archimedean field and let $\overline{K}$ be a separable closure of K. Given an element α in $\overline{K}$, denote its Galois conjugates by α2, ..., αn. Krasner's lemma states:
 * if an element β of $\overline{K}$ is such that
 * $$\left|\alpha-\beta\right|<\left|\alpha-\alpha_i\right|\text{ for }i=2,\dots,n $$
 * then K(α) ⊆ K(β).

Applications

 * Krasner's lemma can be used to show that $\mathfrak{p}$-adic completion and separable closure of global fields commute. In other words, given $$\mathfrak{p}$$ a prime of a global field L, the separable closure of the $$\mathfrak{p}$$-adic completion of L equals the $$\overline{\mathfrak{p}}$$-adic completion of the separable closure of L (where $$\overline{\mathfrak{p}}$$ is a prime of $\overline{L}$ above $$\mathfrak{p}$$).
 * Another application is to proving that Cp &mdash; the completion of the algebraic closure of Qp &mdash; is algebraically closed.

Generalization
Krasner's lemma has the following generalization. Consider a monic polynomial
 * $$f^*=\prod_{k=1}^n(X-\alpha_k^*)$$

of degree n > 1 with coefficients in a Henselian field (K, v) and roots in the algebraic closure $\overline{K}$. Let I and J be two disjoint, non-empty sets with union {1,...,n}. Moreover, consider a polynomial
 * $$g=\prod_{i\in I}(X-\alpha_i)$$

with coefficients and roots in $\overline{K}$. Assume
 * $$\forall i\in I\forall j\in J: v(\alpha_i-\alpha_i^*)>v(\alpha_i^*-\alpha_j^*).$$

Then the coefficients of the polynomials
 * $$g^*:=\prod_{i\in I}(X-\alpha_i^*),\ h^*:=\prod_{j\in J}(X-\alpha_j^*)$$

are contained in the field extension of K generated by the coefficients of g. (The original Krasner's lemma corresponds to the situation where g has degree 1.)