Find a gcd of two polynomials from $\mathbb{Z}[i]$

Question

I'm trying to solve this problem from my abstract algebra course:

Being $z_1=39-8i$ and $z_2=7+i$ elements from the ring of Gauss integers, $\mathbb{Z}[i]=\{a+bi \mid a,b\in\mathbb{Z}\}$. Find a greater common divisor of both terms, $d=\text{gdc}(z_1,z_2)$, and find elements $x_1,x_2\in\mathbb{Z}[i]$ such that $d=x_1z_1 + x_2z_2$.

I have no idea how to proceed with this kind of exercises, I've seen other ones similar to this one in my book and want to find a method to approach these kind of exercises. I decided to post this one in particular because the ring $\mathbb{Z}[i]$ seemed especially weird but interenting, so I assumed this must be one of the most difficult ones of this kind of problems.

How can I solve this? Any help will be appreciated, thanks in advance.

Answer 1

I'd work in the ring $R = \Bbb Z[x]/\langle x^2+1\rangle = \{ax+b\mid a,b\in\Bbb Z\}$, where $x^2+1 = 0$ in $R$.

Then division with rest gives

$-8x+39 = (-8)(x+7) + 95$, i.e., $95 = 1\cdot (-8x+39) + 8\cdot (x+7)$.

Then over the rationals: $1 = \frac{1}{95}(-8x+39) + \frac{8}{95}(x+7)$.