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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.

Lisa
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Mayank Pandey
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  • 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. – Timbuc Oct 30 '14 at 18:59
  • 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. – Lisa Oct 30 '14 at 22:19

1 Answers1

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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.

Dietrich Burde
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