Let be a local ring with maximal ideal . Let be the residue class field, and for a polynomial , let obtained by reduction of the coefficients of modulo . Then is Henselian if, for any monic such that factorizes as with monic and relatively prime in , there exist monic such that with , , and the ideal is the unit ideal in .
Often this definition is given just for the case when one of , is a linear factor , where the idea is that the (simple) root can be lifted to .
A Henselian ring with residue class field is strictly Henselian if is separably closed.
Any field is trivially Henselian.
A complete local ring, such as p-adic number rings, is Henselian. This includes rings of formal power series over a field, .
Rings of convergent power series over a local field are Henselian.
The quotient of a Henselian ring is also Henselian.
In the first place, a quotient of a local ring is local (and has the same residue class field ). Secondly, is clearly Henselian since we can lift factorizations in to factorizations in , and then push them down to .
If is local and its reduced ring (i.e., modulo its ideal of nilpotent elements) is Henselian, then itself is Henselian.
If is Henselian and is a local ring that is integral over (meaning that is an -algebra and each is an integral element over ), then is Henselian.
Let denote the category of local rings (commutative of course) and local ring homomorphisms ( is local if the pullback along of the maximal ideal of is the maximal ideal of ).
The full subcategory of consisting of Henselian rings is reflective. The left adjoint to the full inclusion is called Henselization.
Let be a discrete valuation ring, with its field of fractions. Let be the -adic completion of with respect to its maximal ideal. Then the Henselization of is isomorphic to the subring of whose elements are roots of separable polynomials with coefficients in .
A rule of thumb, as suggested by this example, is that the Henselization is the algebraic part of a local ring completion.
An original reference is
Lecture notes are in
Other sources include
Alonso, Lombardi, Perdry, Henselian local rings (pdf)
Remarks on Henselian rings (pdf);