| 释义 |
Hensel's LemmaAn important result in Valuation Theory which gives information on finding roots of Polynomials.Hensel's lemma is formally stated as follow. Let  be a complete non-Archimedean valuated field, and let  bethe corresponding Valuation Ring. Let  be a Polynomial whose Coefficients are in and suppose  satisfies
 | (1) |
 is the (formal) Derivative of  . Then there exists a unique element  such that  and
 | (2) |
 is``small'' compared to  , then the equation  has a solution ``near'' . In addition, there are no othersolutions near , although there may be other solutions. The proof of the Lemma is based around the Newton-Raphsonmethod and relies on the non-Archimedean nature of the valuation.
Consider the following example in which Hensel's lemma is used to determine that the equation  is solvable inthe 5-adic numbers  (and so we can embed the Gaussian Integers inside in a nice way). Let  be the 5-adic numbers , let  , and let  . Then we have  and  , so
 | (3) |
 such that  and
 | (4) |
 such that  and
 | (5) |
 in . It is possible to find the roots of anyPolynomial using this technique. |