## A fact about irreducible Gaussian integers

Let $\pi \in \mathbb{Z}[i]$ be irreducible which is not real and not an associate of $1+i$. Prove that if $\pi|\alpha$ and $\pi|\overline{\alpha}$, then $N(\pi)|\alpha$.

Since $\pi|\overline{\alpha}$, we have $\pi\tau = \overline{\alpha}$. Note that conjugation preserves products and has order 2; thus $\overline{\pi}\overline{\tau} = \alpha$. Since $\pi|\alpha$ and $\pi$ is irreducible, either $\pi|\overline{\pi}$ or $\pi|\overline{\tau}$. In the first case, since $\pi$ is not real, by this previous exercise, $\pi$ is an associate of $1+i$, a contradiction. So $\pi|\overline{\tau}$. Thus $N(\pi) = \pi\overline{\pi}$ divides $\alpha$.