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 be a separable closure of ''K''. Given an element α in , denote its Galois conjugates by ''α''₂, ..., ''α''_''n''. Krasner's lemma states:
:if an element ''β'' of is such that
::$\left, \alpha-\beta\<\left, \alpha-\alpha_i\\texti=2,\dots,n$
:then ''K''(''α'') âŠ† ''K''(''β'').

Applications

*Krasner's lemma can be used to show that $\mathfrak$-adic completion and separable closure of global fields commute. In other words, given $\mathfrak$ a prime of a global field ''L'', the separable closure of the $\mathfrak$-adic completion of ''L'' equals the $\overline$-adic completion of the separable closure of ''L'' (where $\overline$ is a prime of above $\mathfrak$).
*Another application is to proving that C_p — the completion of the algebraic closure of Q_p — is algebraically closed.

Generalization

Krasner's lemma has the following generalization.Brink (2006), Theorem 6
Consider a monic polynomial
::$f^*=\prod_^n(X-\alpha_k^*)$
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 . Moreover, consider a
polynomial
::$g=\prod_(X-\alpha_i)$
with coefficients and roots in . Assume
::$\forall i\in I\forall j\in J: v(\alpha_i-\alpha_i^*)>v(\alpha_i^*-\alpha_j^*).$
Then the coefficients of the polynomials
::$g^*:=\prod_(X-\alpha_i^*),\ h^*:=\prod_(X-\alpha_j^*)$
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

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*{{Neukirch et al. CNF, edition=2

Lemmas in number theory
Field (mathematics)