Let $(R, \mathfrak m)$ be a Noetherian local domain such that the $\mathfrak m$-adic completion of $R$ is also an integral domain. Let $K$ be the fraction field of $R$ . Let $L$ be a finite extension field of $K$. Let $\overline R_L$ be the integral closure of $R$ in $L$.
Under what conditions on $R$ or $L$ can we say that $\overline R_L$ is also local (I'm especially interested when $L=K$) ? When can we say $\overline R_L$ is also Noetherian ?
I know $\overline R_L$ is a Krull domain by Mori-Nagata theorem and that if $R$ has dimension $1$ , then $\overline R_L$ is Noetherian.
Apart from that I don't know anything else.
Please help.