Show that $\displaystyle \mathbb{Z}[i]$ is a principal ideal domain.
I have already shown that $\displaystyle \mathbb{Z}[i]$ is an integral domain. How do I show that all ideals are principal ($\displaystyle I=(r), r\in \mathbb{Z}[i]$)?
Show that $\displaystyle \mathbb{Z}[i]$ is a principal ideal domain.
I have already shown that $\displaystyle \mathbb{Z}[i]$ is an integral domain. How do I show that all ideals are principal ($\displaystyle I=(r), r\in \mathbb{Z}[i]$)?
Well, then you still can do something that is actually equivalent: show that the norm $\displaystyle N(a+bi):=a^2+b^2$
in $\displaystyle \mathbb{Z}[i]$ permits you to carry on "division with residue" just as the absolute value
allows us to do the same in the integers.
Once you have this proceed as with the integers to show the ring is a PID.
Tonio