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 Zbl 35.0226.02
[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