Find the gcd of the following Gaussian integers

Question

$\gcd(5 + 8i, 3 + 2i)$ in $Z[i]$.

I found it and I got 1 then I look at the manual solution and it turns out it can be i or -i or -1 or 1. why?

Answer 1

It has to do with the definition of a gcd. I am not sure what definition you are using. Possibly you were told that a gcd of two Gaussian integers $u$ and $v$ is any Gaussian integer $z$ that divides both $u$ and $v$, and such that any common divisor of $u$ and $v$ divides $z$.

If we use that definition, one can show that if $z$ is a gcd of $u$ and $v$, and $\epsilon$ is any unit, then $\epsilon z$ is also a gcd of $u$ and $v$. Conversely, if $z\ne 0$ and $z'$ are gcd's of $u$ and $v$, then $z'=\epsilon z$ for some unit $\epsilon$.

So gcd is not unique, it is determined only up to multiplication by a unit. There is no such thing as the gcd.

In the Gaussian integers, the units are $1,-1,i,-i$.