## 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.