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Does there exist a characteristic $0$ principal ideal domain $R$ that has countably infinitely many prime ideals and such that there is no injective unital ring homomorphism $R\rightarrow \overline{\mathbb{Q}}$?

I am aware of examples of PIDs with countably many prime ideals coming from number theory but they are all subrings of $\overline{\mathbb{Q}}$. I am aware of uncountable PIDs like $\mathbb{C}[x]$ (resp. $\mathbb{Z}_p$) but it has uncountably many (resp. finitely many) prime ideals.

user26857
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1 Answers1

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The ring $\Bbb Q[x]$ of polynomials with rational coefficients is a characteristic $0$ PID with countably infinitely many prime ideals, but it is not a subring of $\overline{\Bbb Q}$, as $x$ isn't algebraic over $\Bbb Q$.

user26857
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Arthur
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  • what if we strengthen the condition that there is no injective unital ring homomorphism $R\rightarrow \overline{\mathbb{Q}}$ to the condition that there is no injective unital ring homomorphism $R\rightarrow \mathbb{C}$? Your example does embed in the complex numbers by sending $x$ to something transcendental. –  Aug 19 '19 at 13:12
  • @hello That it does. I don't know, to be honest. – Arthur Aug 19 '19 at 13:13