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.

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