Is a "local enough" class number always equal to one?

Question

Let $F$ be a number field, and $\mathcal{O}$ its ring of integers. Is there always a finite set $S$ of places of $F$ such that $\mathcal{O}_S$ has class number one? Is it a consequence of standard results on class field theory or is it something else?

Answer 1

Yes, it's really just the finiteness of the class number. Let $I$ be a non-principal ideal in $\cal O$. Then $I^h=(a)$ is principal, where $h$ is the classnumber. Let $S_1=\{P_1,\ldots,P_k\}$ be the prime ideals containing $a$, (and so containing $I$). In the Dedekind domain $\mathcal{O}_{S_1}$, the ideal $(a)$ (and so also $I$) become trivial. The class groups of $\mathcal{O}$ surjects onto that of $\mathcal{O}_{S_1}$, and the surjection kills $[I]$. So we get a Dedekind domain with a smaller class number. Now just repeat until the classnumber is $1$.