Is it true that the quadratic ring of integers $O :=\mathcal{O}_{\mathbb{Q}(\sqrt{5})} =\mathbb{Z}(\frac{\sqrt{5}+1}{2})$ is a UFD? How to show it?

My guess: I know that if a quadratic ring of integers is a PID, then it is a UFD. If $O$ has a Dedekine-Hasse norm, then it is a PID (Dummit & Foote p.281), then a UFD. Possibly one may imitate Dummit & Foote p.282 where they show that $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})}$ is a PID, right? Also, this table indicates $O$ has class number 1, I guess it means it is a UFD.

Possibly related: Quadratic rings of integers which are UFD, Real quadratic fields which are UFD (P.s. On algebra, I have only exposure to Dummit & Foote.)