TITLE: how to compute the henselization of some simple rings?
QUESTION [5 upvotes]: Hi,everyone.
I want to know that how to compute the henselization of some simple rings, for example:  $k[x]_{(x)}$ and $R[X]_{(X)}$ where $k$ is a field and $R$ is a excellent DVR.
thank you very much!

REPLY [6 votes]: Perhaps a better source for this result is Nagata's 1962 book, "Local Rings". Chapter VII begins with a section on constructing the henselization of a local ring and its properties. For a normal domain R the above result (note though that the separable algebraic closure, not the algebraic closure must be used in char p) follows since his definition is to take the separable closure of R in the algebraic closure of the function field k(R), intersect it with the fixed field for the decomposition group of the maximal ideal of R, and then localize the resulting subring.