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

Question

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.

Answer 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.