GCD in the Gaussian Integers

Question

How can you prove that if the gcd(a,b) = 1 then gcd(a,bi) = 1 in the Gaussian integers? I know that $i$ is a unit in the ring, but how can you rigorously prove this?

Answer 1

Recall that the gcd is (up to units) the Gaussian integer of smallest norm that can be written as $ma+nb$ for $m,n$ Gaussian integers. If $\gcd(a,b) = 1$, then we have such $m$ and $n$. But then we have $1=ma-in(bi)$, so $\gcd(a,bi)=1$ as well.

Answer 2

When $u$ is a unit, $a$ and $ua$ have the same set of divisors. That's not hard to prove rigorously, and settles the question.