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?