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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?

user26857
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Emre
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  • What? Can you explain the nature of your question? It's weird to ask about this for the ring of algebraic integers rather than the case of a number field's integer ring. – Adam Hughes Apr 21 '16 at 04:57
  • Since going-up theorems don't have a restriction on the finiteness of extension, I wonder if this result can also be extended to the infinite case. – Emre Apr 21 '16 at 05:03
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    Then write $A$ as a direct limit, and note prime ideals as direct limits of prime ideal chains from the finite levels. You can use Chebotarev to ensure enough split primes. – Adam Hughes Apr 21 '16 at 05:15
  • Well, thanks. In general, where can I learn more about this ring? For example, is it Noetherian etc. – Emre Apr 21 '16 at 05:41
  • It's not Notherian, it's actually seldom studied, because the finite cases are so much more interesting. You won't find a lot about the algebraic structure of it because it's just not interesting in that case: the finite case has so much more structure to it. – Adam Hughes Apr 21 '16 at 06:01
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    @AdamHughes I'm not sure it's fair to say that it's not studied or that it's not interesting. Anyone who works with Galois representations will need to understand this ring. Rather I would say that the best (only?) way to study this ring is via the finite case, which makes the finite case a natural starting point. – Mathmo123 Apr 21 '16 at 06:20
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    @Wilf-Fine I'm not sure I understand your last sentence. In a finite extension of $\mathbb Q$, there are only finitely many prime ideals in $\mathcal O_K$ lying over $p$. – Mathmo123 Apr 21 '16 at 06:21
  • Yes, that's what I was trying to say. – Emre Apr 21 '16 at 06:24
  • @Mathmo123 Yes that's what I was getting at, because it's a direct limit it may as well just be a collection of finite things rather than the unified whole; I've at least never heard a compelling argument why you wouldn't just have a perfect understanding of the finite bits and ignore the fact that technically there is a place they all can be unioned together. – Adam Hughes Apr 21 '16 at 21:27

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