Suppose $R$ is a domain, and let $\overline{R}$ denote the integral closure of $R$ in its field of fractions. Does it follow that $\overline{R}[x] \cong \overline{R[x]}$?
We have inclusions $R[x] \rightarrow \overline{R[x]} \rightarrow \text{Frac}(R[x])$, where the last is isomorphic to $\text{Frac}(R)(x).$ Can I use this fact to get what I am looking for? Any guidance is helpful.