nLab Hensel's lemma

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Contents

Idea

The result called Hensel’s lemma is a generalisation of a result due to Kurt Hensel on solving polynomial equations in p-adic number rings. It applies to certain complete topological rings, and now local rings that satisfy the conclusion of the lemma (really a theorem) are called Henselian rings.

Statement

An element xx of a topological ring is called topologically nilpotent if 0 is a limit of the sequence {x n}\{x^n\}. (For example: in any ring AA equipped with a (two-sided) ideal mm, the elements of mm are topologically nilpotent in the mm-adic topology.)

A topological ring is linearly topologized if 0 has a fundamental system of neighbourhoods consisting of ideals.

Theorem

(“Hensel’s Lemma”)

Let AA be a complete Hausdorff linearly topologized commutative ring. Let mm be a closed ideal of AA whose elements are topologically nilpotent. Let B=A/mB=A/m be the quotient topological ring and ϕ:AB\phi\colon A\to B the quotient map. Let RA{X}R\in A\{X\} be a restricted formal power series, P¯\overline{P} a monic polynomial in B[X]B[X] and Q¯B{X}\overline{Q}\in B\{X\}. Suppose that ϕ¯(R)=P¯.Q¯\overline{\phi}(R) = \overline{P}.\overline{Q} and that P¯\overline{P} and Q¯\overline{Q} are strongly relatively prime in B{X}B\{X\}. Then there exists a unique lift of P¯\overline{P} to PA[X]P\in A[X] and of Q¯\overline{Q} to QA{X}Q \in A\{X\} such that R=P.QR=P.Q. Moreover PP and QQ are strongly relatively prime in A{X}A\{X\}, and if RR is a polynomial, so is QQ.

The proof proceeds by considering successively more general statements, starting with various cases in which AA is discrete, in which case RR and Q¯\overline{Q} are polynomials.

Proof

First consider the case that m 2=0m^2=0. Let S,TA[X]S,T\in A[X] with SS monic, ϕ¯(S)=P¯\overline{\phi}(S)=\overline{P} and ϕ¯(T)=Q¯\overline{\phi}(T) = \overline{Q}.

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The second case is that mm is a nilpotent ideal: there is some n2n\geq 2 such that m n=0m^n=0. This is proved by induction on nn, with the base case covered by the first part of the proof.

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The third case assumes merely that AA is discrete, or equivalently that mm is a nil-ideal? (every element in mm is nilpotent, with no global bound on the order). This case considers an ideal nn generated by coefficients of the polynomials at hand, which is then a finitely-generated ideal contained in a nil-ideal, hence nilpotent. We thus can use the second case.

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The last case is the general case, where one considers a fundamental system of open neighbourhoods of 00 by ideals II, whence A/IA/I is discrete.

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For the full proof, see (Bourbaki)

This gives rather simpler looking results in special cases, but all of them boil down to lifting factorisations through a quotient map AA/mA \to A/m.

The original example is of AA being the p-adic integers p\mathbb{Z}_p, with the quotient p𝔽 p= p/p p\mathbb{Z}_p \to \mathbb{F}_p = \mathbb{Z}_p/p\mathbb{Z}_p.

References

The original paper in which a special case of Hensel’s lemma appeared, for monic polynomials over the p-adic integers, is

  • Kurt Hensel, Neue Grundlagen der Arithmetik. Journal für die reine und angewandte Mathematik 127 (1904) 51-84 (EuDML)

and updated to remove monicity in

A proof for more general topological rings is in

  • Bourbaki, Commutative Algebra, III.4.3

See also for simple examples over the pp-adic numbers:

A viewpoint of Hensel’s lemma using étale covers is in Chapitre XI of

  • Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Volume 169 (1970) doi:10.1007/BFb0069571

For an application/quick explanation see this Math.SE answer

A version of Hensel’s lemma for arbitrary continuous functions p p\mathbb{Z}_p \to \mathbb{Z}_p (rather than polynomials or formal power series) is in:

  • Hajime Kaneko, Thomas Stoll, Hensel’s lemma for general continuous functions, arXiv:1707.01445

Last revised on February 28, 2018 at 01:56:34. See the history of this page for a list of all contributions to it.