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 $\mathbb{Z}[i]$.