Hensel lemma

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A statement obtained by K. Hensel [1] 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 [3].


[1] K. Hensel, "Neue Grundlagen der Arithmetik" J. Reine Angew. Math. , 127 (1904) pp. 51–84
[2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[3] H. Zassenhaus, "Ueber eine Verallgemeinerung des Henselschen Lemmas" Arch. Math. (Basel) , 5 (1954) pp. 317–325
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Hensel lemma. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article