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 ($\displaystyle I=(r), r\in \mathbb{Z}[i]$)?

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- Mar 29th 2011, 07:04 PMJJMC89Show that Z[i] is a PID.
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 ($\displaystyle I=(r), r\in \mathbb{Z}[i]$)? - Mar 29th 2011, 07:08 PMtonio
- Mar 29th 2011, 07:10 PMJJMC89
- Mar 29th 2011, 07:14 PMtonio

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