Let $W$ be a finite-dimensional vector space over $% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $. Let $\rho $ $:G\rightarrow $ GL$(W)$ be a representation of any finite group $G$. How can we show that if all characters of $G$ are trivial then $% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion \lbrack W]^{G}$ is a UFD? I know there is a proof in one of Benson's books, but I'm finding it quite hard to follow.
Also, I'm stuck on the following: By considering a representation of $C_{2}$ show that a polynomial invariant ring can fail to be a unique factorisation domain, even in characteristic zero.
Thank you.