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 ${\mathfrak {p}}$ -adic completion 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

^ Lemma 8.1.6 of Neukirch, Schmidt & Wingberg 2008
^ Lorenz (2008) p.78
^ Proposition 8.1.5 of Neukirch, Schmidt & Wingberg 2008
^ Proposition 10.3.2 of Neukirch, Schmidt & Wingberg 2008
^ Lorenz (2008) p.80
^ 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, vol. 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, MR 2392026, Zbl 1136.11001

[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

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