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A statement obtained by K. Hensel [[#References|[1]]] in the creation of the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469301.png" />-adic numbers (cf. [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469302.png" />-adic number]]), which subsequently found extensive use in [[Commutative algebra|commutative algebra]]. One says that Hensel's lemma is valid for a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469303.png" /> with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469304.png" /> if for any unitary polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469305.png" /> and decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469306.png" /> of its reduction modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469307.png" /> into a product of two mutually-prime polynomials
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A statement obtained by K. Hensel [[#References|[1]]] in the creation of the theory of  $  p $-
 +
adic numbers (cf. [[P-adic number| $  p $-
 +
adic number]]), which subsequently found extensive use in [[Commutative algebra|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
 
there exist polynomials
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h0469309.png" /></td> </tr></table>
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$$
 +
Q _ {1} ( X) , Q _ {2} ( X)  \in  A [ X]
 +
$$
  
 
such that
 
such that
  
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$$
 +
P ( X)  = Q _ {1} ( X) \cdot Q _ {2} ( X),\ \
 +
\overline{Q}\; _ {1} ( X)  = q _ {1} ( X),\ \
 +
\overline{Q}\; _ {2} ( X)  = q _ {2} ( X)
 +
$$
  
(here the bar denotes the image under the reduction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693011.png" />). In particular, for any simple root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693012.png" /> of the reduced polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693013.png" /> there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693014.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693015.png" /> which satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693016.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693018.png" />-adic numbers, Hensel's lemma yields the solvability of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693019.png" />, since this equation has two simple roots in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046930/h04693020.png" /> of seven elements. A local ring for which Hensel's lemma is valid is known as a [[Hensel ring|Hensel ring]].
+
(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|Hensel ring]].
  
 
For Hensel's lemma in the non-commutative case see [[#References|[3]]].
 
For Hensel's lemma in the non-commutative case see [[#References|[3]]].

Revision as of 22:10, 5 June 2020


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=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