TITLE: How can one talk about gcd in the context of complex numbers where order doesn't exist?
QUESTION [35 upvotes]: I was reading this: GCD of gaussian integers but the thing that we are calculating, the GCD, doesn't make any sense in the complex number world since there is no ordering of numbers there. So, how can we define and calculate the GCD there?

REPLY [2 votes]: Order does exist, just as it does in $\mathbb{Z}$. For example, what is $\gcd(-28, -98)$? We agree that $14$ is greater than $-14$ even though they are both the same distance away from $0$.
Now try $\gcd(-5 + i, -1 - 3i)$. There are four choices here, we choose to go with the one having positive real part and positive imaginary part.
As it turns out, divisibility provides a way to order numbers. Suppose we have three nonzero, non-unit numbers $a$, $b$, $c$. Then we expect there to be some kind of norm function $N(x)$ such that $N(a) < N(ab) < N(abc)$.