Gaussian integers having relatively prime norms are relatively prime

Let \alpha,\beta \in \mathbb{Z}[i]. Prove that if N(\alpha) and N(\beta) are relatively prime integers, then \alpha and \beta are relatively prime in \mathbb{Z}[i].


Suppose \gamma is a common factor of \alpha and \beta. Say \gamma\xi = \alpha and \gamma\eta = \beta. In particular, note that N(\gamma)|N(\alpha) and N(\gamma)|N(\beta). Since N(\alpha) and N(\beta) are relatively prime, N(\gamma) = 1. Thus \gamma is a unit, and so \alpha and \beta are relatively prime in \mathbb{Z}[i].

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