Hensel’s lemma for integers

Let f(x) be a polynomialMathworldPlanetmathPlanetmath with integer coefficients, p a prime numberMathworldPlanetmath, and n a positive integer.  Assume that an integer a (and naturally its whole residue classMathworldPlanetmath modulo pn) satisfies the congruenceMathworldPlanetmathPlanetmathPlanetmath

f(x) 0(modpn). (1)

The solution  x=a  of (1) may be refined in its residue class modulo pn to a solution  x=a+rpn  of the congruence

f(x) 0(modpn+1). (2)

This refinement is unique modulo pn+1 iff  f(a)0(modp).

Proof.  Now we have  f(a)=spn.  We have to find an r such that

f(a+rpn) 0(modpn+1).

The short Taylor theorem requires that


where  2nn+1, whence this congruence can be simplified to

spn+rf(a)pn 0(modpn+1).

Thus the integer r must satisfy the linear congruence

s+rf(a) 0(modp).

When  f(a)0,  this congruence has a unique solution modulo p (see linear congruence); thus we have the refinement  a=a+rpn  which is unique modulo pn+1.

When  f(a)0  and  s0(modp),  the congruence evidently is impossible.

In the case  f(a)s0(modp)  the congruence (2) is identically true in the residue class of a modulo pn.   □


Title Hensel’s lemma for integers
Canonical name HenselsLemmaForIntegers
Date of creation 2013-04-08 19:35:26
Last modified on 2013-04-08 19:35:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition
Classification msc 11A07