Let $p(x)\in \mathbb{Z}[x]$ be a monic, irreducible polynomial in $\mathbb{Z}[x]$. For the field $K:=\mathbb{Q}[x]/(p(x))$, its ring of algebraic integers $\mathcal{O}_K$ is always a Dedekind domain trivially.
Since the ring $R:=\mathbb{Z}[x]/(p(x))$ is a subring of $\mathcal{O}_K$ (and sometimes they are even identical), I am curious about properties of $R=\mathbb{Z}[x]/(p(x))$.
For instance, is $R=\mathbb{Z}[x]/(p(x))$ a Dedekind domain as well?
Clearly, $R$ is an integral domain, and is noetherian, too. But I am unable to prove/refute (1), $R$ is integrally closed, and (2) every nonzero prime ideal of $R$ is maximal.
Can we prove that $R$ is an integral domain? Or can we impose some strong restrictions on $p(x)$ such that the resultant $R$ must be a Dedekind domain?
Thanks!!
Just to expand on Mathmo123's comment above, did you consider any examples, e.g. $p(x) = x^2 + 3$?
Incidentally, you should be able to prove that any non-zero prime ideal is maximal.