Henselian ring

Summary

A Henselian ring is a local ring in which Hensel's lemma holds. There, factorizations of polynomials over the residue field into coprime factors lift to the ring. Every complete local ring, such as the $p$-adic integers or a formal power series ring, is Henselian. Henselian rings behave like small neighbourhoods of a point in the étale topology. Every local ring has a Henselization, a smallest Henselian ring over it.