Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of ${\mathbb{Q}}$ (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial.

Definition 1 A Dedekind domain is a Noetherian integral domain ${A}$ that is integrally closed, and of Krull dimension one—that is, each nonzero prime ideal is maximal.   (more…)