Is there a commutative integral domain $R$ in which:
- every nonzero prime ideal $Q$ is maximal, and
- there are maximal ideals $Q$ with $R/Q$ of sizes $3$, $11$, and $27$?
This doesn't happen with number rings of Galois extensions of $\mathbb{Q}$, as far as I know, since what happens to $3$ seriously happens to $3$. You cannot have residue fields of sizes both $3$ and $27$. Now $(\mathbb{Z}/3\mathbb{Z})[X]$ has residue fields of sizes $3$ and $27$, but not $11$.
Edit: I'll separate the next into its own question.