symmetric monoidal (∞,1)-category of spectra
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.
An element of a topological ring is called topologically nilpotent if 0 is a limit of the sequence . (For example: in any ring equipped with a (two-sided) ideal , the elements of are topologically nilpotent in the -adic topology.)
A topological ring is linearly topologized if 0 has a fundamental system of neighbourhoods consisting of ideals.
(“Hensel’s Lemma”)
Let be a complete Hausdorff linearly topologized commutative ring. Let be a closed ideal of whose elements are topologically nilpotent. Let be the quotient topological ring and the quotient map. Let be a restricted formal power series, a monic polynomial in and . Suppose that and that and are strongly relatively prime in . Then there exists a unique lift of to and of to such that . Moreover and are strongly relatively prime in , and if is a polynomial, so is .
The proof proceeds by considering successively more general statements, starting with various cases in which is discrete, in which case and are polynomials.
First consider the case that . Let with monic, and .
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The second case is that is a nilpotent ideal: there is some such that . This is proved by induction on , with the base case covered by the first part of the proof.
\ldots
The third case assumes merely that is discrete, or equivalently that is a nil-ideal? (every element in is nilpotent, with no global bound on the order). This case considers an ideal 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 by ideals , whence 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 .
The original example is of being the p-adic integers , with the quotient .
The original paper in which a special case of Hensel’s lemma appeared, for monic polynomials over the p-adic integers, is
and updated to remove monicity in
A proof for more general topological rings is in
See also for simple examples over the -adic numbers:
A viewpoint of Hensel’s lemma using étale covers is in Chapitre XI of
For an application/quick explanation see this Math.SE answer
A version of Hensel’s lemma for arbitrary continuous functions (rather than polynomials or formal power series) is in:
Last revised on February 28, 2018 at 01:56:34. See the history of this page for a list of all contributions to it.