Is there a commutative integral domain $R$ in which:
- every nonzero prime ideal $Q$ is maximal, and
- for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such that $R/Q$ is a field of size $q$?
I am looking for a ring where "$q\equiv 3 \bmod 8$" describes finite fields, rather than just finite local rings like in the ring $\mathbb{Z}$. The only examples I can think of that satisfy the first condition have a finite number of residue fields of each characteristic. The only examples I can think of that satisfy the second condition have a ton of non-maximal non-proper prime ideals with all sorts of bizarre fields associated to them.