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!!