Krasner’s lemma

Krasner’s lemma (along with Hensel’s lemma) connects valuationsPlanetmathPlanetmath on fields to the algebraic structurePlanetmathPlanetmath of the fields, and in particular to polynomial roots.

Lemma 1.

(Krasner’s Lemma) Let K be a field of characteristic 0 completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with respect to a nontrivial nonarchimedean absolute valueMathworldPlanetmathPlanetmath. Assume α,βK¯ (where K¯ is some algebraic closureMathworldPlanetmath of K) are such that for all nonidentity embeddingsMathworldPlanetmathPlanetmathPlanetmath σHomK(K(α),K¯) we have |α-β|<|σ(α)-α|. Then K(α)K(β).

This says that for any αK¯, there is a neighborhoodMathworldPlanetmathPlanetmath of α each of whose elements generates at least the same field as α does.


It suffices to show that for every σHomK(β)(K(α,β),K¯), we have σ(α)=α, for then α is in the fixed field of every embedding of K(β), so αK(β). Note that


where the final equality follows since |σ()| is another absolute value extending ||K to K(α,β) and thus must be equal to ||. But then


But this is impossible by the bounds on α,β unless σ(α)=α. ∎

The first application of Krasner’s lemma is to show that splitting fieldsMathworldPlanetmath are “locally constant” in the sense that sufficiently close polynomialsMathworldPlanetmathPlanetmathPlanetmath in K[X] have the same splitting fields.

Proposition 2.

With K as above, let P(X)K[X] be a monic irreducible polynomialMathworldPlanetmath of degree n with (distinct) roots α1,,αn. Then any monic polynomialMathworldPlanetmath Q(X)K[X] of degree n that is “sufficiently close” to P(X) will be irreduciblePlanetmathPlanetmath over K with roots β1,βn, and (after renumbering) K(αi)=K(βi).

Here “sufficiently close” means the following: consider the space of degree n polynomials over K as homeomorphic to Kn as a topological spaceMathworldPlanetmath; close then means close in the obvious metric induced by ||.


Since P(X) has distinct roots, we may choose 0<γ<min(|αi-αj|) for ijn. Since the roots of a polynomial vary continuously with its coefficients, we say that a degree n polynomial Q(X)K[X] is sufficiently close to P(X) if Q(X) has roots β1,,βn with |αi-βi|<γ. But {αj}ji are all the Galois conjugates of αi, and |αi-βi|<γ<|αi-αj| by construction, so by Krasner’s lemma, K(αi)K(βi). But


so that K(βi)=K(αi). In additionPlanetmathPlanetmath, we see that degQ=[K(βi):K] and thus that Q(X) is irreducible. ∎

We use this fact to show that every finite extensionMathworldPlanetmath of p arises as a completion of some number fieldMathworldPlanetmath.

Corollary 3.

Let K be a finite extension of Qp of degree n. Then there is a number field E and an absolute value || on E such that E^K.


Let K=p(α) and let P be the minimal polynomial for α over p. Since is dense in p, we can choose Q(X)[X] (note: in [X], not p[X]), and β a root of Q(X), as in the propositionPlanetmathPlanetmath, so that p(α)=p(β). Let E=(β). Clearly E is a number field which, when regarded as embedded in p(β), has absolute value ||E, the restrictionPlanetmathPlanetmathPlanetmathPlanetmath of the absolute value on p(α)=p(β). Then E^ is a complete field with respect to that absolute value; p(β) is as well, and E is dense in both, so we must have E^=p(β)=p(α)=K. ∎

Finally, we can prove the following generalizationPlanetmathPlanetmath of Krasner’s Lemma, which is also given that name in the literature:

Lemma 4.

Let K be a field of characteristic 0 complete with respect to a nontrivial nonarchimedean absolute value, and K¯ an algebraic closure of K. Extend the absolute value on K to K¯; this extensionPlanetmathPlanetmathPlanetmath is unique. Let K¯^ be the completion of K¯ with respect to this absolute value. Then K¯^ is algebraically closed.


Let α be algebraic over K¯^ and P(X) its monic irreducible polynomial in K¯^[X]. Since K¯ is dense in K¯^, by proposition 2 we may choose Q(x)K¯[X] with a root βK¯^ such that K¯^(α)=K¯^(β). But K¯^(β)=K¯^ so that αK¯^. ∎

Title Krasner’s lemma
Canonical name KrasnersLemma
Date of creation 2013-03-22 19:03:02
Last modified on 2013-03-22 19:03:02
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 12J99
Classification msc 11S99
Classification msc 13H99