## In the Gaussian integers, the conjugate of a prime is prime

Prove that if $a+bi$ is irreducible in $\mathbb{Z}[i]$, then so is $b+ai$.

Note that conjugation preserves multiplication in $\mathbb{Z}[i]$; $\overline{\alpha\beta} = \overline{\alpha} \overline{\beta}$ for all $\alpha$ and $\beta$. As a consequence, if $\alpha$ is irreducible then $\overline{\alpha}$ is as well.

Note that $b+ai = \overline{(-i)(a+bi)}$. Thus, if $a+bi$ is irreducible, then so is $b+ai$.