TITLE: Given a group $G$, does there exist a domain $D$ with $G$ as its ideal class group?
QUESTION [8 upvotes]: I have only recently encountered algebraic number theory and was wondering if this is the case. If the answer to the question is yes, then can we explicitly construct the domain $D$ ?

REPLY [8 votes]: See Claborn, Luther (1966), "Every abelian group is a class group", Pacific J. Math. 18: 219–222. You can get it here