Talk:Euclidean domain

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Small Confusion

At the top of the article, it says that Euclidean domains are a superset of fields, but in the examples for Euclidean domain, it says "any field". But wouldn't this mean that these terms equivalent...? 128.146.164.146 (talk) 00:05, 23 February 2012 (UTC)[reply]

Every field is a Euclidean domain, not every Euclidean domain is a field. A field (any field) is an example of a Euclidean domain. I don't see the problem. Pirround (talk) 15:45, 20 April 2012 (UTC)[reply]

PIDs vs EDs in second paragraph

The second paragraph is really about the difference between PIDs and EDs for which a Euclidean function is given and there is additionally an algorithm for computing q and r. Otherwise, it's not so clear that EDs are any more concrete than PIDs, as neither come with explicit algorithms for instantiating Bezout's identity. It would be nice if the language of the paragraph made this more explicit. — Preceding unsigned comment added by 99.37.200.120 (talk) 17:24, 18 November 2013 (UTC)[reply]

I have corrected the paragraph to remove "concreteness", which is an editor's opinion, and thus has not its place in WP. I have also corrected the implicit assertion that GCD is computable in ED. In fact, GCD and Bézout's identity are easily computable as soon as one has an algorithm for Euclidean division (that is an algorithm for the quotient). But for most Euclidean domains the computation of the quotient is not easy. For Euclidean domains that occur in number theory, when the Euclidean function is the square root of the norm, Euclidean division amounts to find the closest vector in a lattice, which is a difficult problem related to the lattice problem and effective Minkowski's theorem. For more general Euclidean functions, the problem is much more dificult. D.Lazard (talk) 19:07, 18 November 2013 (UTC)[reply]

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]