Hensel’s lemma

The following results are used to show the existence of a solution to polynomial equations over local fieldsMathworldPlanetmath. Notice the similarities with Newton’s method.

Theorem (Hensel’s Lemma).

Let K be a local field (http://planetmath.org/LocalField), completePlanetmathPlanetmathPlanetmathPlanetmath with respect to a valuationMathworldPlanetmathPlanetmath ||. Let OK be the ring of integersMathworldPlanetmath in K (i.e. the set of elements of K with |k|1). Let f(x) be a polynomialMathworldPlanetmathPlanetmathPlanetmath with coefficients in OK and suppose there exist α0OK such that


Then there exist a root αK of f(x). Moreover, the sequenceMathworldPlanetmath:


converges to α. Furthermore:

Corollary (Trivial case of Hensel’s lemma).

Let K be a number fieldMathworldPlanetmath and let p be a prime idealPlanetmathPlanetmath in the ring of integers OK. Let Kp be the completion of K at the finite place p and let Op be the ring of integers in Kp. Let f(x) be a polynomial with coefficients in Op and suppose there exist α0Op such that


Then there exist a root αKp of f(x), i.e. f(α)=0.

Title Hensel’s lemma
Canonical name HenselsLemma
Date of creation 2013-03-22 15:08:30
Last modified on 2013-03-22 15:08:30
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 13H99
Classification msc 12J99
Classification msc 11S99