# how to compute the henselization of some simple rings?

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!

• I have the impression that in the first case, you can identify it with the subring of formal power series $p\in k[[x]]$ which are algebraic in the sense that $f(p)=0$ for some polynomial $f$. – Donu Arapura Feb 10 '12 at 9:48

In both cases, the henselization $A^h$ of your ring $A$ is its algebraic closure in its completion $\hat{A}$.This follows for instance from Artin approximation (Algebraic approximation of structures over complete local rings, Publications Mathématiques de l'IHÉS, 36, 1969, p. 23-58): any system of polynomial equations that has a solution in $\hat{A}$ has a solution in $A^h$ (and you may find a solution as close as you want to the original one).