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
(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 .
|||K. Hensel, "Neue Grundlagen der Arithmetik" J. Reine Angew. Math. , 127 (1904) pp. 51–84|
|||N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)|
|||H. Zassenhaus, "Ueber eine Verallgemeinerung des Henselschen Lemmas" Arch. Math. (Basel) , 5 (1954) pp. 317–325|
Hensel lemma. V.I. Danilov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hensel_lemma&oldid=15048