Is there a quicker way to work out highest common factors in $\Bbb{Z}[i]$?

Question

I wanted to find HCF of $5+i$ and $3+5i$.

So I considered norms and found that the factors of the units in $\Bbb{Z}[i]$, and the associates of $5+i$ for the first. And units in $\Bbb{Z}[i]$, the associates of $1+i$, the associates of $4+i$, and the associates of $3+5i$ in the second.

So the only common factors are the units.

So I concluded that HCF was $\{\pm1,\pm i\}$.

However this approach to quite a while of working and I wondered if there was a smarter approach rather than just directly calculating the factors in each case and comparing?

Thanks!

Answer 1

Use the Euclidean algorithm for Gaussian Integers.