Is the number of prime ideals in a ring of algebraic integers countable?

Question

My question is: If $K$ is a number field, then $Spec(\mathcal O_K)$ is countable or non-countable.

Answer 1

Recall that:

• The number of finite subsets of a countable set is countable.

• Every ideal is finitely generated in this case.

From this you should be able to derive the fact that the number of prime ideals, and indeed of all ideals, is countable in this case.