# Hensel lemma

A statement obtained by K. Hensel [1] in the creation of the theory of $p$- adic numbers (cf. $p$- adic number), which subsequently found extensive use in commutative algebra. One says that Hensel's lemma is valid for a local ring $A$ with maximal ideal $\mathfrak m$ if for any unitary polynomial $P( X) \in A[ X]$ and decomposition $\overline{P}\; ( X) = q _ {1} ( X) \cdot q _ {2} ( X)$ of its reduction modulo $\mathfrak m$ into a product of two mutually-prime polynomials

$$q _ {1} ( X) , q _ {2} ( X) \in ( A/ \mathfrak m ) [ X] ,$$

there exist polynomials

$$Q _ {1} ( X) , Q _ {2} ( X) \in A [ X]$$

such that

$$P ( X) = Q _ {1} ( X) \cdot Q _ {2} ( X),\ \ \overline{Q}\; _ {1} ( X) = q _ {1} ( X),\ \ \overline{Q}\; _ {2} ( X) = q _ {2} ( X)$$

(here the bar denotes the image under the reduction $A \rightarrow A/ \mathfrak m$). In particular, for any simple root $\alpha$ of the reduced polynomial $\overline{P}\; ( X)$ there exists a solution $a \in A$ of the equation $P( X) = 0$ which satisfies the condition $\overline{a}\; = \alpha$. 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 $\mathbf Z _ {7}$ of $7$- adic numbers, Hensel's lemma yields the solvability of the equation $X ^ {2} - 2 = 0$, since this equation has two simple roots in the field $\mathbf F _ {7}$ 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].

#### References

 [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
How to Cite This Entry:
Hensel lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hensel_lemma&oldid=47210
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article