nLab
Hensel's lemma

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}.

\ldots

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.

\ldots

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.

\ldots

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.

\ldots

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 27, 2018 at 20:56:34. See the history of this page for a list of all contributions to it.