# Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID

How would one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a principal ideal domain (PID)? It isn't a Euclidean domain according to the Wikipedia article on PIDs.

• Read the last example in sec. 8.2 in Dummit&Foote's "Abstract Algebra" . It all boils down to proving that the corresponding field norm is a Dedekind-Hasse norm on this ring and thus it is a PID. – Timbuc Oct 30 '14 at 18:59
• Ah, very interesting. If you could prove that $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})}$ is a Euclidean domain then you don't have to prove that it is a principal ideal domain (PID) because that follows automatically from its being a Euclidean domain. – Lisa Oct 30 '14 at 22:19