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.

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