Why is the ring $\mathbb{Z}[2\sqrt{2}]$ not a unique factorization domain?

Question

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

I'm trying to find an irreducible non-prime element or something but I don't know where to start.

Answer 1

It is enough to show that $\mathbb{Z}[2\sqrt{2}]$ is not a unique factorisation domain (why?).

The elements $2$ and $2\sqrt{2}$ are irreducible and $$ 8 = (2\sqrt{2})^2 = 2^3, $$ so the factorisation is not unique.

Answer 2

Hint $ $ Over $\, R = \Bbb Z[w],\ w =2\sqrt{2},\,$ the proper fraction $\,w/2\,$ is a root of monic $\,x^2-2,\,$ contra the Rational Root Test (which is true in any UFD; i.e. UFDs are integrally closed).