I know that $I:=\{a+ib \in \mathbb Z[i] : 2\mid a-b\}$ is a maximal ideal of $\mathbb Z[i]$. My question is: what is the number of elements of $\mathbb Z[i] /I $?
I am totally stuck. Please help .
(Since $\mathbb Z[i]$ is a euclidean domain, $I$ must be generated by some element $\alpha \in \mathbb Z[i]$, then $|\mathbb Z[i] /I|=|\alpha|^2$ but I have no idea what this $\alpha$ should be here)
If $z=a+bi$ does not fulfill $2\mid a-b$, then $z-1$ does. Hence the order of the quotient is $2$. (The generator of the ideal is $1+i$).