Let $k$ be a field. Consider the Laurent polynomial ring $R=k[x_1,x_1^{-1},\ldots ,x_n,x_n^{-1}]$. I need to show that $R$ is integrally closed. For $n=1$ I have proved that it will be integrally closed as localisation at every prime ideal will give a DVR and hence a integrally closed ring.
I need some help to prove for $n>1$.
Thank you!
Have you not yet proved that localization preserves integral closure? That's easily found, e.g. here: https://proofwiki.org/wiki/Localization_Preserves_Integral_Closure
$R=k[x_1,x_2,...,x_n]$ is a UFD, hence integrally closed, and the Laurent polynomials are just a localization of that.