# Thread: Show that Z[i] is a PID.

1. ## Show that Z[i] is a PID.

Show that $\mathbb{Z}[i]$ is a principal ideal domain.

I have already shown that $\mathbb{Z}[i]$ is an integral domain. How do I show that all ideals are principal ( $I=(r), r\in \mathbb{Z}[i]$)?

2. Originally Posted by JJMC89
Show that $\mathbb{Z}[i]$ is a principal ideal domain.

I have already shown that $\mathbb{Z}[i]$ is an integral domain. How do I show that all ideals are principal ( $I=(r), r\in \mathbb{Z}[i]$)?

If you know, or can show by means of the norm, that that ring is an Euclidean one then you're done, otherwise I

can't see how to show directly that it is a PID...

Tonio

3. Originally Posted by tonio
If you know, or can show by means of the norm, that that ring is an Euclidean one then you're done, otherwise I can't see how to show directly that it is a PID...
I haven't covered Euclidean domains or norms.

4. Originally Posted by JJMC89
I haven't covered Euclidean domains.

Well, then you still can do something that is actually equivalent: show that the norm $N(a+bi):=a^2+b^2$

in $\mathbb{Z}[i]$ 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