Let $K$ be an arbitrary number field. I know that $\mathcal{O}_K$ is a Dedekind domain. I know that it need not be a PID. However, despite a long search, I am having a hard time finding either an affirmative or negative answer to the question of whether $\mathcal{O}_K$ is necessarily a GCD domain. If anyone knows the answer to this question, then please let me know.
I also have a related question: Let $S$ be any finite set of places of $K$ that contains all the Archimedian places. Let $\mathcal{O}_{K, S}$ be the ring of $S$-integers of $K$; that is, the set of all $x \in K$ such that for each place $v \notin S,$ we have $|x|_v \leq 1.$ Is $\mathcal{O}_{K, S}$ necessarily a GCD domain?
Thank you very much for your attention.
Best Answer
A Dedekind domain $D$ is Noetherian so atomic, i.e. nonunits $\neq 0$ factor into atoms = irreducibles.
Thus $D$ is a GCD domain $\iff D$ is a UFD, by the same proof as in $\,\Bbb Z,\,$ i.e. gcds exist therefore Euclid's Lemma holds, which implies atoms are prime.
$D$ Dedekind $\Rightarrow D$ is $1$-dimensional (prime ideals $\!\neq\! 0$ are max), thus $D$ is a UFD $\!\iff\! D$ is a PID.
Hence a Dedekind domain $D$ is a GCD domain $\iff D$ is a UFD $\iff D$ is a PID.