TITLE: Why is not the ring $\mathbb{Z}[2\sqrt{2}]$ a unique factorization domain?
QUESTION [8 upvotes]: Why is not the polynomial ring $R[x]$ a unique factorization domain, where $R$ is the quadratic integer ring $\mathbb{Z}[2\sqrt{2}]$? 

I'm trying to find a irreducible nonprime element or something but I don't know where to start.

REPLY [2 votes]: Hint $\ R = \Bbb Z[w],\ w =2\sqrt{2}$ is not integrally closed since $\,w/2\not\in R\,$ is a root of $\,x^2-2.\,$ But any UFD is integrally closed (monic case of the Rational Root Test).