Henselian ring

In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings are , , and . In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings. A local ring R with maximal ideal m is called Henselian if Hensel's lemma holds. This means that if P is a monic polynomial in R[x], then any factorization of its image P in (R/m)[x] into a product of coprime monic polynomials can be lifted to a factorization in R[x]. A local ring is Henselian if and only if every finite ring extension is a product of local rings. A Henselian local ring is called strictly Henselian if its residue field is separably closed. By abuse of terminology, a field with valuation is said to be Henselian if its valuation ring is Henselian. That is the case if and only if extends uniquely to every finite extension of (resp. to every finite separable extension of , resp. to , resp. to ). A ring is called Henselian if it is a direct product of a finite number of Henselian local rings. Assume that is an Henselian field. Then every algebraic extension of is henselian (by the fourth definition above). If is a Henselian field and is algebraic over , then for every conjugate of over , . This follows from the fourth definition, and from the fact that for every K-automorphism of , is an extension of . The converse of this assertion also holds, because for a normal field extension , the extensions of to are known to be conjugated. Henselian rings are the local rings with respect to the Nisnevich topology in the sense that if is a Henselian local ring, and is a Nisnevich covering of , then one of the is an isomorphism. This should be compared to the fact that for any Zariski open covering of the spectrum of a local ring , one of the is an isomorphism.