Krasner's lemma

Template:Short description 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 Template:Overline be a separable closure of K. Given an element α in Template:Overline, denote its Galois conjugates by α₂, ..., α_n. Krasner's lemma states:

if an element β of Template:Overline is such that

$|\alpha -\beta |<|\alpha -{\alpha }_i|\text{ for }i=2,\dots ,n$

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 $\overline{𝔭}$-adic completion of the separable closure of L (where $\overline{𝔭}$ is a prime of Template:Overline above $𝔭$).
• 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. Consider a monic polynomial

$f^{*}=\prod _{k=1}^n(X-{\alpha }_k^{*})$

of degree n > 1 with coefficients in a Henselian field (K, v) and roots in the algebraic closure Template:Overline. Let I and J be two disjoint, non-empty sets with union {1,...,n}. Moreover, consider a polynomial

$g=\prod _{i\in I}(X-{\alpha }_i)$

with coefficients and roots in Template:Overline. Assume

$\mathrm{\forall }i\in I\mathrm{\forall }j\in J:v({\alpha }_i-{\alpha }_i^{*})>v({\alpha }_i^{*}-{\alpha }_j^{*}).$

Then the coefficients of the polynomials

$g^{*}:=\prod _{i\in I}(X-{\alpha }_i^{*}),h^{*}:=\prod _{j\in J}(X-{\alpha }_j^{*})$

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.)

References

• Template:Cite journal
• Template:Cite book
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• Template:Neukirch et al. CNF