## Hensel's Lemma

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 non-Archimedean 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 Newton-Raphson method 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 in the 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)

and the condition is satisfied. Hensel's lemma then tells us that there is a 5-adic number such that and
 (4)

Similarly, there is a 5-adic 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.