TITLE: Example of an excellent Henselian regular local ring containing a field that is not the formal power series ring
QUESTION [5 upvotes]: I am reading this paper
http://arxiv.org/pdf/0709.3628.pdf
and I was trying to construct examples of excellent Henselian regular local rings containing a field that are not complete, but could not come up with any. Can someone enlighten with an example?

REPLY [4 votes]: Inside the ring $k[[ x]]$, consider the subring $R$ of elements which are algebraic over $k(x)$. The ring $R$ is Henselian because the Henselian condition is all about equations being solvable, so you can always satisfy the Henselian condition with algebraic elements. I leave it to you to see that $R$ is a regular local ring (of dimension $1$) and is not complete.