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

abstract-algebra unique-factorization-domains

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.


Solution 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.

Solution 2:

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).

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