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 be a separable closure of K. Given an element α in, denote its Galois conjugates by α₂, ..., α_n. Krasner's lemma states:

Applications

Krasner's lemma can be used to show that -adic completion and separable closure of global fields commute. In other words, given a prime of a global field L, the separable closure of the -adic completion of L equals the -adic completion of the separable closure of L.
Another application is to proving that C_p — the completion of the algebraic closure of Q_p — is algebraically closed.

Generalization

Krasner's lemma has the following generalization.
Consider a monic polynomial
of degree n > 1
with coefficients in a Henselian field and roots in the
algebraic closure. Let I and J be two disjoint,
non-empty sets with union. Moreover, consider a
polynomial
with coefficients and roots in. Assume
Then the coefficients of the polynomials
are contained in the field extension of K generated by the
coefficients of g.