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
where is the (formal) Derivative of . Then there exists a unique element such that and
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
and the condition is satisfied. Hensel's lemma then tells us that there is a
5-adic number such that
Similarly, there is a 5-adic number such that and
Therefore, we have found both the square roots of in . It is possible to find the roots of any
Polynomial using this technique.
© 1996-9 Eric W. Weisstein