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, ..., αn. Krasner's lemma states:Lemma 8.1.6 of Lorenz (2008) p.78
Applications Krasner's lemma can be used to show that -adic completion and separable closure of global fields commute.Proposition 8.1.5 of 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 Cp—the completion of the algebraic closure of Qp—is algebraically closed.Proposition 10.3.2 of Lorenz (2008) p.80
Generalization Krasner's lemma has the following generalization.Brink (2006), Theorem 6 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.)
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