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

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

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