I've been stuck on this problem for a while now.

Suppose $\displaystyle 32 = \alpha\beta$ for $\displaystyle \alpha,\beta$ relatively prime quadratic integers in $\displaystyle \mathbb{Q}[i]$. Show that $\displaystyle \alpha=\epsilon\gamma^{2}$ for some unit $\displaystyle \epsilon$ and some quadratic integer $\displaystyle \gamma$ in $\displaystyle \mathbb{Q}[i]$.

Any pointers on how to start?

Thanks.