View source for Hensel lemma

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

<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/h0469308.png" /></td> </tr></table>

there exist polynomials

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such that

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(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]].

For Hensel's lemma in the non-commutative case see [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Hensel,   "Neue Grundlagen der Arithmetik"  ''J. Reine Angew. Math.'' , '''127'''  (1904)  pp. 51–84</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Zassenhaus,   "Ueber eine Verallgemeinerung des Henselschen Lemmas"  ''Arch. Math. (Basel)'' , '''5'''  (1954)  pp. 317–325</TD></TR></table>