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, ..., α_n. Krasner's lemma states:^[1][2]

if an element β of K is such that

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

then K(α) ⊆ K(β).

Applications

• Krasner's lemma can be used to show that $\mathfrak{p}$ and separable closure of global fields commute.^[3] In other words, given $\mathfrak{p}$ a prime of a global field L, the separable closure of the $\mathfrak{p}$-adic completion of L equals the $\overline{\mathfrak{p}}$-adic completion of the separable closure of L (where $\overline{\mathfrak{p}}$ is a prime of L above $\mathfrak{p}$).
• Another application is to proving that C_p — the completion of the algebraic closure of Q_p — is algebraically closed.^[4][5]

Generalization

Krasner's lemma has the following generalization.^[6] 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 K. 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 K. 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

$g^*:=\prod_{i\in I}(X-\alpha_i^*),\ h^*:=\prod_{j\in J}(X-\alpha_j^*)$

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.)

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, Zbl 1136.11001, MR 2392026