Let $A$ be the ring of all elements of $\mathbb{C}$ that are integral over $\mathbb{Z}$, and $p\in\mathbb{Z}$ a prime element. Are there infinitely many prime ideals of $A$ lying over $p\mathbb{Z}$?
I know this conclusion is true for a number field, i.e. when we change $\mathbb{C}$ with a finite extension of $\mathbb{Q}$. But, I can't answer this question?