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.

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