Show that $\mathbb{Z}[\sqrt{-2}]$ is unique factorization domain.

Question

Show that $\mathbb{Z}[\sqrt{-2}]$ is unique factorization domain. I am unable to proceed. Any help, thanks in advance.

Answer 1

Show that it is Euclidean, using the norm. The essential point is that every element in the ambient $\mathbb C$ can be translated by an element of $\mathbb Z[\sqrt{-2}]$ to a point in the rectangle $R$ where $|\Im(z)|\le \sqrt{2}/2$ and $|\Re(z)|\le 1/2$. The points of largest norms are the corners, with norms $\sqrt{3/4}<1$.

By this inequality, given a Gaussian integer $\alpha$ and another non-zero Gaussian integer $\beta$, we want a Gaussian integer $q$ and remainder $r$ such that $\alpha=q\beta+r$, with $|r|<|\beta|$. Dividing through by $\beta$, this asks that, given $\alpha/\beta$, can we find a Gaussian integer $q$ such that $|{\alpha\over \beta}-q|<1$. The previous estimate assures that this is possible.