Does there exist a $1$-dimensional Noetherian domain whose set of all prime ideals has cardinality $\aleph_1$ ?

Question

Does there exist a $1$-dimensional Noetherian domain whose set of all prime ideals has cardinality $\aleph_1$ ?

What I know is that for $1$-dimensional Noetherian domain $R$, $\operatorname{max} \operatorname {Spec} R$, under Zariski subspace topology of $\operatorname{Spec} R$, is actually the co-finite topology. But I don't think that would help implying existence of any such domain with specified cardinality of prime spectrum.

Answer 1

Let $K$ be an algebraically closed field. Then the polynomial ring $K[X]$ has $|K|$ prime ideals, namely $\{0\}$ and the $\left<X-a\right>$ for $a\in K$. There is an algebraically closed field of each infinite cardinality.