An 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 nonArchimedean valuated field, and let be
the corresponding Valuation Ring. Let be a Polynomial whose Coefficients are in
and suppose satisfies

(1) 
where is the (formal) Derivative of . Then there exists a unique element such that and

(2) 
Less formally, if is a Polynomial with ``Integer'' Coefficients and is
``small'' compared to , then the equation has a solution ``near'' . In addition, there are no other
solutions near , although there may be other solutions. The proof of the Lemma is based around the NewtonRaphson
method and relies on the nonArchimedean nature of the valuation.
Consider the following example in which Hensel's lemma is used to determine that the equation is solvable in
the 5adic numbers (and so we can embed the Gaussian Integers inside
in a nice way). Let be the 5adic numbers , let , and let . Then we have
and , so

(3) 
and the condition is satisfied. Hensel's lemma then tells us that there is a
5adic number such that
and

(4) 
Similarly, there is a 5adic number such that and

(5) 
Therefore, we have found both the square roots of in . It is possible to find the roots of any
Polynomial using this technique.
© 19969 Eric W. Weisstein
19990525