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, ..., ÃŽÂ±_n. Krasner's lemma states:

if an element β of is such that

:

then K(α) Ã¢ÂŠÂ† K(β).

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 (where is a prime of above ).
• 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 (K, v) and roots in the algebraic closure . Let I and J be two disjoint, non-empty sets with union {1,...,n}. 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. (The original Krasner's lemma corresponds to the situation where g has degree 1.)

Notes

References