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Krasner's lemma

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Relates the topology of a complete non-archimedean field to its algebraic extensions

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, ..., αn. Krasner's lemma states:

if an element β of K is such that
| α − β | < | α − α i |  for  i = 2 , … , n {\displaystyle \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 p {\displaystyle {\mathfrak {p}}} -adic completion and separable closure of global fields commute. In other words, given p {\displaystyle {\mathfrak {p}}} a prime of a global field L, the separable closure of the p {\displaystyle {\mathfrak {p}}} -adic completion of L equals the p ¯ {\displaystyle {\overline {\mathfrak {p}}}} -adic completion of the separable closure of L (where p ¯ {\displaystyle {\overline {\mathfrak {p}}}} is a prime of L above p {\displaystyle {\mathfrak {p}}} ).
  • Another application is to proving that Cp — the completion of the algebraic closure of Qp — is algebraically closed.

Generalization

Krasner's lemma has the following generalization. Consider a monic polynomial

f ∗ = ∏ k = 1 n ( X − α k ∗ ) {\displaystyle 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 = ∏ i ∈ I ( X − α i ) {\displaystyle g=\prod _{i\in I}(X-\alpha _{i})}

with coefficients and roots in K. Assume

∀ i ∈ I ∀ j ∈ J : v ( α i − α i ∗ ) > v ( α i ∗ − α j ∗ ) . {\displaystyle \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 ∗ := ∏ i ∈ I ( X − α i ∗ ) ,   h ∗ := ∏ j ∈ J ( X − α j ∗ ) {\displaystyle 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

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