Puneet Varma (Editor)

Krasner's lemma

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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.

Contents

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:

if an element β of K is such that | α − β | < | α − α i |  for  i = 2 , … , n then K(α) ⊆ K(β).

Applications

  • Krasner's lemma can be used to show that p -adic completion and separable closure of global fields commute. In other words, given p a prime of a global field L, the separable closure of the p -adic completion of L equals the p ¯ -adic completion of the separable closure of L (where p ¯ is a prime of L above 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

    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.)

    References

    Krasner's lemma Wikipedia
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