Take the tour ×  First I would like to be clear about the
definition, which I am having trouble finding.  What does: The
local ring $A$ is algebraically closed in $B\supset A$. (e.g.
for $B:=\hat{A}$, the completion of $A$ at its maximal ideal).
1- Does it mean that every element of $B$ satisfying a
polynomial with coefficients in $A$, belongs to $A$?  2- Does it
mean the same as above but assuming $b$ is an element of some
commutative finite extension of $B$?  Afterwards I am trying to
understand why if $A$ is local, noetherian, Henselian, and
$\hat{A}$ is normal, characteristic is zero, then $A$ is
algebraically closed in $\hat{A}$.  share|improve this question
Crossposted from math.SE: math.stackexchange.com/q/425539/264 – 
Zev Chonoles Jun 20 at 20:55 I suppose $A$ is noetherian. If
$\widehat{A}$ is normal then it is a domain, so $A$ is a domain.
Normality descends through a flat local map of local noetherian
rings (Cor. to Theorem 23.9 in Matsumura "Commutative ring
theory"), so $A$ is normal. Hence, we