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.
Let K be a complete non-archimedean field and let K be a separable closure of K. Given an element α in K, denote its Galois conjugates by α2, ..., αn.
Krasner's lemma states:[1][2] Krasner's lemma has the following generalization.
[6] Consider a monic polynomial of degree n > 1 with coefficients in a Henselian field (K, v) and roots in the algebraic closure K. Let I and J be two disjoint, non-empty sets with union {1,...,n}.
Moreover, consider a polynomial with coefficients and roots in K. 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.)