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.
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 $\Bbb Q$.