9
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Timbuc Oct 30 '14 at 18:59
  • $\begingroup$ 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. $\endgroup$ – Lisa Oct 30 '14 at 22:19
9
$\begingroup$

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.) An example of a PID that is not a Euclidean Domain.

3.) A principal ideal domain that is not Euclidean.

4.) On a Principal Ideal Domain that is not a Euclidean Domain.

5.) Ring of integers is a PID but not a Euclidean domain.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.