Gaussian integers which are relatively prime need not have relatively prime norms

Exhibit two Gaussian integers \alpha and \beta which are relatively prime in \mathbb{Z}[i] but such that N(\alpha) and N(\beta) are not relatively prime integers.

Note that 1+2i and 1-2i have norm 5. Using this previous exercise, 1+2i and 1-2i do not divide each other, and thus are relatively prime. In particular, 1+2i = (-1+i)(1-2i) + (-i). However, these elements both have norm 5, and so their norms are not relatively prime.

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