I'm looking for an easy proof for that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID.
One proof I know is to show that the field norm is a Dedekind-Hasse norm, but this proof is quite dirty( that it proves case by case) and technical.
Since any ring is a PID iff it admits a Dedekind-Hasse norm, I think every proof would look similar to the one I know. However, since this problem has been on qualifying exam quite several times , I'm curious if there is one which is less technical. Thank you in advance.