Talk:Euclidean domain

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Things which are not euclidean rings

It would be nice to list some examples of some things that are NOT euclidean rings. I'm not an algebraist and it's been a very long time since I studied such things, so I was looking back to determine if the polynomials in several variable was a euclidean ring. — Preceding unsigned comment added by 98.155.236.135 (talk) 06:36, 22 May 2014 (UTC)[reply]

Done D.Lazard (talk) 10:41, 22 May 2014 (UTC)[reply]

Codomain of euclidean function

I think it should be the naturals union the zero element, and not just the naturals as the article says, because in that case for the polinomials over a field the degree of a constant polinomial should be zero, or the degree shouldn't be an euclidean function. — Preceding unsigned comment added by 181.29.18.118 (talk) 18:13, 1 December 2014 (UTC)[reply]

Please, place the new sections at the end of the talk page and sign your posts with four tildes (~~~~).

In mathematics, the natural numbers commonly include zero. Nevertheless, I have edited the article for clarification. D.Lazard (talk) 18:32, 1 December 2014 (UTC)[reply]

Multiplicative Euclidean functions

The last paragraph of section § Definition defines the concept of a multiplicative Euclidean function without any evidence that this concept is commonly considered. In view of the high number of textbooks that consider Euclidean domains, a reference to a textbook is required to establish the notability of the concept. Wikipedia is an encyclopedia, not a database for all definitions that have been ever given. Also, if the concept would be notable, it would have been studied which Euclidean rings of integers of algebraic number fields have a multiplicative Ruclidean funcion. Indeed, as mentioned in the article, most rings of integers of a number field that are principal ideal domains are Euclidean (possibly under a generalized Riemann hypothesis) and the norm is a Euclidean function for only very few examples. The fact that multiplicative Euclidean functions are generally not mentionad in this contex suggests that it is unknown whether these non-norm Euclidean functions are multiplicative or not, and that the multiplicative property is not important.

So, I suggest to remove this paragraph unless a source is provided for the notability of the concept.

Recently a user insists to add to this paragraph the fact that somebody proved that there are Euclidean domains which do not have a multiplicative Euclidean function. This is WO:OR, since the source is a WP:primary source that seems to not have been mentioned in any textbook. So, unless reliable secondary source are provided, this is WP:original research, and therefore not suitable for Wikipedia. In any case such a property does not belong to section § Definition.

So, I'll revert again the recent addition, and wait for a consensus for removing the definition of multiplicative Euclidean functions. D.Lazard (talk) 14:57, 29 May 2024 (UTC)[reply]

Non-Euclidean rings of integers of number fields

It should be mentioned that, if a number field K has class number 1, then O_K is a ED other than $\mathbb {Q} ({\sqrt {-19}}),\mathbb {Q} ({\sqrt {-43}}),\mathbb {Q} ({\sqrt {-67}}),\mathbb {Q} ({\sqrt {-163}})$ . This is, for example, mentioned here.

For the ED but not norm-ED case $\mathbb {Z} \left[{\dfrac {1+{\sqrt {69}}}{2}}\right]$ , the explicit Euclidean function given in the paper should be shown in the article. It is also mentioned here. 129.104.65.7 (talk) 08:17, 20 November 2024 (UTC)[reply]

The cited paper proves your first assertion only if some conjectures are assumed to be true. A similar result assuming a generalized Riemann hypothesis was proved by Weinsberger a long time ago. For quadratic integers, this is reported in Quadratic integer#Euclidean rings of quadratic integers.

My opinion is that these results do not belong to this article, but Weinberger's theorem sould be mentioned in Algebraic integer. The new result that uses other conjectures is a WP:primary source that is not notable enough for being mentioned here (Scholar Google says that this article is cited only 8 times, and only twice by regularly published papers) D.Lazard (talk) 09:28, 20 November 2024 (UTC)[reply]

Real quadratic field

The ring of integers of a real quadratic field with class number 1 and discriminant < 500 is proved to be Euclidean; see Section 5.2, Page 34, Theorem 6 of here. Of course this concerns $\mathbb {Z} [{\sqrt {14}}],\mathbb {Z} [{\sqrt {22}}],\mathbb {Z} [{\sqrt {23}}],\cdots$ . Note still that $\mathbb {Z} \left[{\dfrac {1+{\sqrt {69}}}{2}}\right]$ is current the only case where an explicit Euclidean function is constructed; for other rings in the family we just say that the "minimal Euclidean function" is well-defined. 129.104.65.7 (talk) 08:27, 20 November 2024 (UTC)[reply]