## Use the Dedekind-Hasse criterion to show that ZZ[i] is a unique factorization domain

Use the Dedekind-Hasse criterion to show that $\mathbb{Z}[i]$ is a unique factorization domain.

Let $\alpha,\beta \in \mathbb{Z}[i]$ be nonzero such that $\beta$ does not divide $\alpha$ and $N(\alpha) > N(\beta)$. By the division algorithm in $\mathbb{Z}[i]$, there exist $\gamma,\delta$ such that $\alpha = \gamma\beta + \delta$ and $0 < N(\delta) < N(\beta)$. So $0 < N(\alpha - \gamma\beta) < N(\delta)$. So $\mathbb{Z}[i]$ satisfies the Dedekind-Hasse criterion, and thus is a unique factorization domain.

Note that this is a somewhat silly way to prove this result; the division algorithm makes $\mathbb{Z}[i]$ a Euclidean domain and thus a unique factorization domain via some more general theorems.