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.
Solution 1:
This is a classical example. Here are a few references (out of many) which give a detailed proof.
1.) An example of a PID that is not a Euclidean Domain.
2.) A principal ideal domain that is not Euclidean.
3.) On a Principal Ideal Domain that is not a Euclidean Domain.
4.) Ring of integers is a PID but not a Euclidean domain.
5.) An example of a PID that is not a Euclidean Domain.