Krasner's lemma explained

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 α₂, ..., α_n. Krasner's lemma states:^{[1] [2]}

if an element β of is such that

\left|\alpha - \beta\right|<\left|\alpha - \alpha _{i\right|fori=2,...,n}

then K(α) ⊆ K(β).

Applications

• Krasner's lemma can be used to show that ak{p} -adic completion and separable closure of global fields commute.^[3] In other words, given

ak{p} a prime of a global field L, the separable closure of the ak{p} -adic completion of L equals the \overline{ ak{p}} -adic completion of the separable closure of L (where \overline{ ak{p}} is a prime of above ak{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 ^{*)} _{k} of degree n > 1with coefficients in a Henselian field (K, v) and roots in thealgebraic closure . Let I and J be two disjoint,non-empty sets with union . Moreover, consider apolynomial g = \prod _{i\in} (X - \alpha _{i)} with coefficients and roots in . Assume\forall i\in I\forall j\in J: v(\alpha _{i-\alpha} ^{*).} ^{j} Then the coefficients of the polynomialsg ^{*: = \prod} _{i\in I} (X - \alpha ^{*), h} _{i} ^{*: = \prod} ^{j\in J} (X - \alpha ^{*)} _{j} are contained in the field extension of K generated by thecoefficients of g. (The original Krasner's lemma corresponds to the situation where g has degree 1.)

Notes

1. Lemma 8.1.6 of
2. Lorenz (2008) p.78
3. Proposition 8.1.5 of
4. Proposition 10.3.2 of
5. Lorenz (2008) p.80
6. Brink (2006), Theorem 6

References

• Brink . David . New light on Hensel's Lemma . 24 . 4 . Expositiones Mathematicae . 291–306 . 2006 . 0723-0869 . 1142.12304 . 10.1016/j.exmath.2006.01.002.
• Book: Lorenz, Falko . Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics . 2008 . . 978-0-387-72487-4 . 1130.12001 .
• Book: Narkiewicz, Władysław . Elementary and analytic theory of algebraic numbers . 3rd . 1159.11039 . Springer Monographs in Mathematics . Berlin . . 3-540-21902-1 . 2004 . 206 .