Studying about normalizations I've bumped in the following theorem:
Theorem. Let $R$ be a normal (integrally closed) domain, then $R[x]$ is a normal domain.
How to prove (elegantly, if possible) it?
Is true that if $R$ a normal domain, then $R[[x]]$ is a normal domain? How to prove it?
Thank you for help.
EDIT Now it's clear that, if $R$ is normal $R[[x]]$ is not necessary normal. In the answer, Martin Brandenburg cited the condition of be "completely integrally closed" and a counter example with DVR of dimension equal to 2.
What about if $R$ is a PID or in general a ring of dimension 1?