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I am reading this paper

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?

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up vote 4 down vote accepted

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.

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Thank you, David. I don't think I would have come up with something like this on my own. – Sasukaro T Oct 5 '12 at 3:58
Just noted the adjective "excellent" above. Assuming $char(k)=0$, this is a characteristic zero Dedekind domain, so we're OK. I assume we are also OK in finite characteristic, but I was never good at working with the excellent condition. – David Speyer Oct 10 '12 at 19:40

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