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I am looking for an example of a quadratic field $\mathbb Q[\sqrt d]$ , with $d \equiv 2 $ or $3\pmod 4$ , whose ring of integers is Euclidean but not norm Euclidean.

Please help. Thanks in advance.

user26857
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  • This is incomplete: http://oeis.org/wiki/Euclidean_domain The ones with question marks are where I would (or I will) start looking. –  Apr 14 '15 at 13:24
  • @JyrkiLahtonen : That is not much helpful , I am asking for Euclidean evaluations which are not field norms ( i.e. not even absolute value of field norms ) where as the thread concentrates on real Euclidean domains not valuations –  Apr 14 '15 at 13:31
  • You're right, Saun Dev. I was fairly sure that an example has been posted here (or at MO) earlier. I've been looking for it for about 8 minutes. That was just a placeholder :-) My recollection was that $d=23$ is a counterexample, and my searching had reached this, when Gerry's answer came. – Jyrki Lahtonen Apr 14 '15 at 13:34
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    Saun Dev, I found a better local link. – Jyrki Lahtonen Apr 14 '15 at 13:39
  • @JyrkiLahtonen : Thanks , the wiki link there in the first answer really helped , I finally got a positive answer that imaginary quadratic field , if is Euclidean , then is norm Euclidean . –  Apr 14 '15 at 13:43
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    The boilerplate "your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." really doesn't apply to a question like this, which is more along the lines of a reference request than a do-my homework-for-me quesiton. I'm voting to reopen. – Gerry Myerson Jan 03 '17 at 02:38

1 Answers1

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See M. Harper, ${\bf Z}[\sqrt{14}]$ is Euclidean, Canad. J. Math. Vol. 56 (2004), 55–70, available here. It's also done in Bernhard Lutzmann, Quadratic number fields that are Euclidean but not norm-Euclidean, available here. Also worth a look is Malcolm Harper and M. Ram Murty, Euclidean rings of algebraic integers, Canad. J. Math. Vol. 56 (1), 2004 pp. 71–76, available here.

user26857
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Gerry Myerson
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    @SaunDev This is a good answer but an even better answer would give a Euclidean domain and its associated Euclidean function. See this question from a couple of months ago: http://math.stackexchange.com/questions/1148364/what-is-the-euclidean-function-for-mathbbz-sqrt14 The only answer so far says this ring is Euclidean but not norm-Euclidean and that its Euclidean function can be obtained. Apparently it's way more involved than the simple adjustment to the norm required for $\mathcal{O}_{\mathbb{Q}(\sqrt{69})}$ (yeah, I know $69 \equiv 1 \bmod 4$). – Robert Soupe Apr 15 '15 at 00:50