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.

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