Hensel’s Lemma

New light on Hensel’s lemma

The historical development of Hensel’s lemma is briefly discussed (section 1). Using Newton polygons, a simple proof of a general Hensel’s lemma for separable polynomials over Henselian fields is given (section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (section 4) and continuity of factors (section 6) are demonstrated. Using this ...

Multivariate Hensel’s Lemma for Complete Rings

In this set of notes, we prove that a complete ring satisfies the multivariate Hensel’s lemma (Theorem 1.11). The result can be seen as a formal version of the implicit function theorem. We make this connection precise in §3. The section on completion follows the Stacks project [4], since most textbooks on this subject assumes Noetherian property. Our exposition of Hensel’s lemma largely follow...

Maps on ultrametric spaces, Hensel’s Lemma, and differential equations over valued fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel’s Lemma and the multidimensional Hensel’s Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions...

Products Of Quadratic Polynomials With Roots Modulo Any Integer

We classify products of three quadratic polynomials, each irreducible over Q, which are solvable modulo m for every integer m > 1 but have no roots over the rational numbers. Polynomials with this property are known as intersective polynomials. We use Hensel’s Lemma and a refined version of Hensel’s Lemma to complete the proof. Mathematics Subject Classification: 11R09

Hensel’s lemma, valuations, and p-adic numbers