nLab Hensel's lemma

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

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

Theorem

(“Hensel’s Lemma”)

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

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

Proof

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

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

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The third case assumes merely that $A$ is discrete, or equivalently that $m$ is a nil-ideal? (every element in $m$ is nilpotent, with no global bound on the order). This case considers an ideal $n$ 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 $0$ by ideals $I$, whence $A/I$ 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 $A \to A/m$.

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

Related concepts

• Henselian ring
• Henselian pair

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

• Kurt Hensel, Theorie der algebraischen Zahlen, 1908. (Internet Archive)

A proof for more general topological rings is in

• Bourbaki, Commutative Algebra, III.4.3

See also for simple examples over the $p$-adic numbers:

• Wikipedia, Hensel’s lemma

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 $\mathbb{Z}_p \to \mathbb{Z}_p$ (rather than polynomials or formal power series) is in:

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