Q:

In general how to prove or disprove certain types of ideal?

i've come  across a lot of questions recently that ask you whether or not there exist certain kinds of ideal, say;
does there exist an ideal$ J $of $\mathbb{Z}[i]$ for which $\mathbb{Z}[i] /J$ is a field of 8 elements?
or for $R ={\{a + bi\sqrt3: a,b \in Z}\} $show there is an ideal $I$ of $R$ so that $R/I$ is isomorphic to $\mathbb{Z}_7,$ but no ideal so that $R/J$ is isomorphic to $\mathbb{Z}_5$?
i am struggling to find examples in such questions, and also in proving them impossible. Is there any intuition to know if such things exist? and any standard approach to answering these questions with ideals? An approach to begin with?

A:

For the case of $\, \Bbb Z[i]\,$: try to prove the nice
Claim: For any $\,a+bi\in\Bbb Z[i]\,\,,\,\,\gcd(a,b)=1\,$ , we have that
$$\Bbb Z[i]/\langle a+bi\rangle\,\cong\Bbb Z/(a^2+b^2)\Bbb Z=:\Bbb Z_{a^2+b^2}$$
Assuming the above, the question is now whether we can write $\,8=a^2+b^2\,,\,a,b\in\Bbb Z\,,\,(a,b)=1$ , which we can't...
Could you do something similar with $\,\Bbb Z[\sqrt{-3}]\,$ ?

