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 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]

if an element β of K is such that
then K(α)  K(β).

Applications

  • Krasner's lemma can be used to show that -adic completion and separable closure of global fields commute.[3] 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 L above ).
  • Another application is to proving that Cp the completion of the algebraic closure of Qp is algebraically closed.[4][5]

Generalization

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.)

Notes

  1. Lemma 8.1.6 of Neukirch, Schmidt & Wingberg 2008
  2. Lorenz (2008) p.78
  3. Proposition 8.1.5 of Neukirch, Schmidt & Wingberg 2008
  4. Proposition 10.3.2 of Neukirch, Schmidt & Wingberg 2008
  5. Lorenz (2008) p.80
  6. Brink (2006), Theorem 6

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, 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, MR 2392026, Zbl 1136.11001
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