# Ideals of the Gaussian Integers

• Feb 9th 2011, 12:40 PM
matt.qmar
Ideals of the Gaussian Integers
Hello!

Given the Gaussian Integers $\displaystyle Z[i] = \{ a + bi | a,b \in Z\}$

Then I am trying to show that

1) $\displaystyle I = \{ a + bi\ |\ a,b \in Z, 5|a, 5|b\}$ is an ideal in $\displaystyle Z[i]$ but $\displaystyle I$ is not maximal. To be clear, I mean 5 divides a, and 5 divides b.

To show it's an ideal, I have to show that it "absorbs" any element from Z[i] "into"
it... seems like something that should follow from number theory. Also, I am not sure how it is not maximal! I need to find some other ideal that I is properly contained in? I can't think of any!

2) $\displaystyle J = <2+ i>$ is NOT a prime ideal in $\displaystyle Z[i]$ (<> refers to the principal ideal)...

and then I am trying to figure out how many elements there are in the factor ring $\displaystyle \frac{Z[i]}{J}$

Any guidance or assistance appreciated! Thanks!!
• Feb 9th 2011, 01:15 PM
roninpro
For part one, you have to show that for any element $\displaystyle z=x+yi\in \mathbb{Z}[i]$, $\displaystyle zI\subseteq I$. Did you try multiplying $\displaystyle (x+yi)(a+bi)$, and checking the condition?

To finish that problem, try showing that $\displaystyle I\subset \langle 1+i\rangle$.

For part two, if you can decide what $\displaystyle \mathbb{Z}[i]/J$ is, then you can resolve your question about the primality of $\displaystyle J$. (Recall that an ideal $\displaystyle I$ is a prime ideal if and only if $\displaystyle R/I$ is an integral domain.)