I'm trying to find the GCD of $(85,1+13i)$ and $(47-13i,53+56i)$. I've tried, but to no avail. I keep setting it up and trying to do it with the same mindset as if i'm doing polynomial division, is this incorrect? I believe I need to do "synthetic division", is that correct? Yeah, if somebody could help me out that'd be great
2026-03-25 07:47:17.1774424837
Need help with Euclidean Algorithm in $\mathbb{Z}[i]$
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$$ \frac{85}{1+13i} = \frac{85}{1+13i} \frac{1-13i}{1-13i}\\ = \frac{85 - 85*13 i}{1+169} = \frac{85}{170} + \frac{-1105}{170}\\ = (0 - 7 i) + (\frac{85}{170} + \frac{1}{2}i)\\ (1+13i)(-7i) = 91-7i\\ 85 - (91 - 7i) = -6 + 7i\\ GCD(85,1+13i) = GCD(85-(1+13i)(-7i),1+13i)\\ =GCD(-6+7i,1+13i) $$
Keep going like this. I suggest writing a script to do this automatically as part of the exercise.