Does there exist an integral domain $R$ of characteristic $p > 0$ that is perfect (i.e., $x \mapsto x^p$ is bijective on $R$) but not integrally closed in its field of fractions?



A perfect domain that is not integrally closed?
Let $S$ be the local ring of nodal curve, $R$ = inverse limit $Frob: S \to S$. For example:


* $k$ a perfect field,
* $f(x,y)=y^{p+1}-x^{p+1}(1+x)$,
* $R=k[x^{1/p^{\infty}},y^{1/p^{\infty}}]/(f^{1/p^{\infty}})$.


Here's a complete local example:


* $k$ a perfect field,
* $f(x,y)=y^{p}-x^{p}y-x^{p+1}$,
* $R=k[[x^{1/p^{\infty}},y^{1/p^{\infty}}]]/(f^{1/p^{\infty}})$.


In each example $(y/x)$ is integral over $R$.


