We want to find a maximal ideal of the ring $\mathbb{Z}[2i]$. We guess that the ideal $I=(5,2i)$ generated by $5, 2i$ is a maximal ideal. All things I know about the ring are Noetherian, but not PID, UFD, Dedekind domain. How can we obtain the form of maximal ideals of $\mathbb{Z}[2i]$?
Thanks for any hints on this problem.
Best Answer
If you accept that $\mathbb Z[2i]$ is isomorphic to $\mathbb Z[X]/(X^2+4)$, then let us find a maximal ideal in the later ring. The maximal ideal $(2,X)$ of $\mathbb Z[X]$ contains $X^2+4$, so $(2,X)/(X^2+4)$ is a maximal ideal in the factor ring. It corresponds to the ideal $(2,2i)$ in $\mathbb Z[2i]$.