## Find the common divisors of two Gaussian integers

FInd all of the nonunit, nonassociate, and nonconjugate common divisors of $\alpha = 9+3i$ and $\beta = -3+7i$ in $\mathbb{Z}[i]$.

Note that $9+3i = 3(3+i)$. Since 3 is irreducible and does not divide $-3+7i$, we have $\mathsf{gcd}(9+3i, -3+7i) = \mathsf{gcd}(3+i, -3+7i)$. Note that $3+i = (1+i)(2-i)$, and that these factors are irreducible since their norms are prime. Now $-3+7i = (1+i)(2+5i)$, and these factors are also irreducible. Thus the greatest common divisor of $\alpha$ and $\beta$ is $1+i$. Since this element is irreducible, it is (up to associates) the only nontrivial common divisor of $\alpha$ and $\beta$.