A Simpler proof for a Principal Ideal Domain which is not a Euclidean Domain

Asked Sep 14 '17 at 00:26

Active Sep 14 '17 at 04:22

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It is well know that $\mathbb{Z} [(1 + \sqrt{-19})/2]$ is a PID which is not a Euclidean Domain. It is also accepted to be the simplest ring for which this is true.

I am wondering if anyone knows of a ring with this property that may be more complicated in its definition but the proof that it is a PID and not a ED is shorter. It is okay if this example uses ideas beyond basic ring theory.

edited Sep 14 '17 at 04:22

asked Sep 14 '17 at 00:26

edenstar

1

Presumably, you mean $\frac{1+\sqrt{-19}}{2}$? – Thomas Andrews Sep 14 '17 at 00:29
Because I tend to understand things better that way. Also I am looking for a shorter proof which will take less time to write on my qualifying exam :) – edenstar Sep 14 '17 at 04:21
Thanks Thomas Andrews I fixed it. – edenstar Sep 14 '17 at 04:22
2

This article and this MSE answer might be helpful. (No doubt you are already aware of "simple" proofs for $\mathbb{Z}[(1+\sqrt{-19})/2]$ such as this article and its references, and of course Keith Conrad's always-great exposition.) – Zach Teitler Sep 14 '17 at 06:34

A Simpler proof for a Principal Ideal Domain which is not a Euclidean Domain

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