# Hensel lemma

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A statement obtained by K. Hensel  in the creation of the theory of -adic numbers (cf. -adic number), which subsequently found extensive use in commutative algebra. One says that Hensel's lemma is valid for a local ring with maximal ideal if for any unitary polynomial and decomposition of its reduction modulo into a product of two mutually-prime polynomials there exist polynomials such that (here the bar denotes the image under the reduction ). In particular, for any simple root of the reduced polynomial there exists a solution of the equation which satisfies the condition . Hensel's lemma is fulfilled, for example, for a complete local ring. Hensel's lemma makes it possible to reduce the solution of an algebraic equation over a complete local ring to the solution of the corresponding equation over its residue field. Thus, in the ring of -adic numbers, Hensel's lemma yields the solvability of the equation , since this equation has two simple roots in the field of seven elements. A local ring for which Hensel's lemma is valid is known as a Hensel ring.

For Hensel's lemma in the non-commutative case see .

How to Cite This Entry:
Hensel lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hensel_lemma&oldid=15048
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article