examples for Hensel’s lemma

Example 1.

Let p be a prime numberMathworldPlanetmath greater than 2. Are there solutions to x2+7=0 in the field p (the p-adic numbers (http://planetmath.org/PAdicIntegers))? If there are, -7 must be a quadratic residueMathworldPlanetmath modulo p. Thus, let p be a prime such that


where (p) is the Legendre symbolDlmfMathworldPlanetmath. Hence, there exist α0 such that α02-7modp. We claim that x2+7=0 has a solution in p if and only if -7 is a quadratic residue modulo p. Indeed, if we let f(x)=x2+7 (so f(x)=2x), the element α0p satisfies the conditions of the (trivial case of) Hensel’s lemma. Therefore there exist a root αp of x2+7=0.

Example 2.

Let p=2. Are there any solutions to x2+7=0 in 2? Notice that if we let f(x)=x2+7, then f(x)=2x and for any α02, the number f(α0)=2α0 is congruentMathworldPlanetmath to 0 modulo 2. Thus, we cannot use the trivial case of Hensel’s lemma.

Let α0=12. Notice that f(1)=8 and f(1)=2. Thus


and the general case of Hensel’s lemma applies. Hence, there exist a 2-adic solution to x2+7=0. The following is the 2-adic canonical form (http://planetmath.org/PAdicCanonicalForm) for one of the solutions:

Title examples for Hensel’s lemma
Canonical name ExamplesForHenselsLemma
Date of creation 2013-03-22 15:08:36
Last modified on 2013-03-22 15:08:36
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Example
Classification msc 12J99
Classification msc 11S99
Classification msc 13H99