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 K be a separable closure of K. Given an element α in K, denote its Galois conjugates by α2, ..., αn. Krasner's lemma states:
Applications Krasner's lemma can be used to show that p {\displaystyle {\mathfrak {p}}} -adic completion and separable closure of global fields commute. In other words, given p {\displaystyle {\mathfrak {p}}} a prime of a global field L, the separable closure of the p {\displaystyle {\mathfrak {p}}} -adic completion of L equals the p ¯ {\displaystyle {\overline {\mathfrak {p}}}} -adic completion of the separable closure of L (where p ¯ {\displaystyle {\overline {\mathfrak {p}}}} is a prime of L above p {\displaystyle {\mathfrak {p}}} ). Another application is to proving that Cp—the completion of the algebraic closure of Qp—is algebraically closed.
Generalization Krasner's lemma has the following generalization. Consider a monic polynomial
f ∗ = ∏ k = 1 n ( X − α k ∗ ) {\displaystyle 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 K. Let I and J be two disjoint, non-empty sets with union {1,...,n}. Moreover, consider a polynomial
g = ∏ i ∈ I ( X − α i ) {\displaystyle g=\prod _{i\in I}(X-\alpha _{i})}
with coefficients and roots in K. Assume
∀ i ∈ I ∀ j ∈ J : v ( α i − α i ∗ ) > v ( α i ∗ − α j ∗ ) . {\displaystyle \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 ∗ := ∏ i ∈ I ( X − α i ∗ ) , h ∗ := ∏ j ∈ J ( X − α j ∗ ) {\displaystyle 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.)
Notes
References Brink, David (2006). "New light on Hensel's Lemma". Expositiones Mathematicae. 24 (4): 291–306. doi:10.1016/j.exmath.2006.01.002. ISSN 0723-0869. Zbl 1142.12304. Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer-Verlag. ISBN 978-0-387-72487-4. Zbl 1130.12001. Narkiewicz, Władysław (2004). Elementary and analytic theory of algebraic numbers. Springer Monographs in Mathematics (3rd ed.). Berlin: Springer-Verlag. p. 206. ISBN 3-540-21902-1. Zbl 1159.11039. Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, MR 2392026, Zbl 1136.11001
