TITLE: Is the formal power series ring integrally closed?
QUESTION [10 upvotes]: Let $k$ be a field and $s$ and $t$ be variables.
Is the ring $k[s][[t]]$ integrally closed in $k[s,s^{-1}][[t]]$?

REPLY [17 votes]: No. Let $\ell$ be a prime invertible in $k$ and consider 
$$x= s (1+ t/s)^{1/\ell} = s  + \frac{t }{\ell} - \frac{(\ell-1) t^2}{ 2s \ell^2} +  \frac{ (\ell-1) (2\ell-1) t^3}{ 6 s^2 \ell^3} + \dots  \in  k[s,s^{-1}][[t]] $$
Clearly it does not lie in $k[s][[t]]$. But we have $$x^\ell = s^{\ell} (1+t/s) = s^\ell + s^{\ell-1} t$$ so it satisfies a monic polynomial equation over $k[s][[t]]$ (and even $k[s,t]$).