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Can anyone give an example of a principal ideal domain that is not Euclidean and is not isomorphic to $\mathbb{Z}[\frac{1+\sqrt{-a}}{2}]$, $a = 19,43,67,163$?

I believe it is conjectured that no other integer rings of number fields have this property. What about other rings?

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One such simple example of a non-Euclidean PID is $ K[[x,y]][1/(x^2\!+\!y^3)]\,$ over any field $\,K,\,$ i.e. adjoin the inverse of $\,x^2\!+\!y^3$ to a bivariate power series ring over a field. For the proof, and a general construction method see

D.D. Anderson. An existence theorem for non-euclidean PID’s,
Communications in Algebra, 16:6, 1221-1229, 1988.

For number rings, by Weinberger (1973), assuming GRH, a UFD number ring R with infinitely many units is Euclidean, e.g. real quadratic number rings are Euclidean $\!\iff\!$ PID $\!\iff\!$ UFD.

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  • $\begingroup$ This is a great reference! $\endgroup$ – NotfromBrazil May 28 '13 at 20:38
  • $\begingroup$ Might I ask for a free pdf online? Or is there any other free way of reading the article? In any case, still thanks in advance. $\endgroup$ – awllower May 29 '13 at 11:28
  • $\begingroup$ @awllower I don't have access at the moment (above is from notes). Probably someone in chat can help. $\endgroup$ – Key Ideas May 30 '13 at 13:24
  • $\begingroup$ OK. No problem with that. And thanks for replying. $\endgroup$ – awllower May 30 '13 at 13:41
  • $\begingroup$ @KeyIdeas : If you do have access to a proof of the result, it would be really appreciated! 39 euros for this is robbery, considering that the journal is not the author and they're getting the money.. I'm guessing the author gets nothing after 28 years. Heh.. $\endgroup$ – Patrick Da Silva Jun 3 '16 at 13:09

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