I'm trying to determine whether $\mathbb{Z}[X]/(X^2+1)$ is an ID, UFD, PID, field.
My progress so far: I know that field $\Rightarrow$ PID $\Rightarrow$ UFD $\Rightarrow$ ID. The units of the ring are $\pm 1$, $\pm X$ (by considering $(aX+b)(cX+d)=1$, equating coefficients and using that $X^2 + 1 = 0$). So the ring isn't a field as not every element is a unit.
How should I continue?
First show that the ring is isomorphic to the ring $\mathbb{Z}[i]$ of Gaussian integers. Then find out here at MSE what is known about this ring (quite a lot is known and has been answered. To give an example, see here). In fact, it is norm Euclidean, hence a PID and hence an UFD. You have shown that $\mathbb{Z}[i]$ is not a field, and as a subring of $\mathbb{C}$ it is clearly a domain, i.e., ID.
