Talk:Euclidean domain/Archive 1

Archive 1

Comments

I felt that the information provided by 128.40.56.75 was worth keeping, if it's correct. Yes, it was poorly formatted, so I've made a first stab at doing better. I've tried to format the references consistently, and I hope I haven't changed them semantically. I have not verified that the actual information provided by 128.40.56.75 is correct, but it sounds very reasonable, so I have no reason to doubt it. Adam1729 03:27, 19 February 2007 (UTC)

I feel like the inequality should stay in the defintion. This inequality isn't obvious for the gaussian integers for example. stephane — Preceding unsigned comment added by 24.16.72.51 (talk) 10:53, 12 August 2007 (UTC)

The paper "Kenneth Rogers: The Axioms for Euclidean Domains" in AMM [1] compares various definitions of Euclidean domains -- N(a)<=N(a)N(b); N(ab)=N(a)N(b) and "the range of N is subset of Z, bounded from bellow". This could be added as a reference since it is related to the stuff in the article. (Perhaps also the text could be reformulated using this article, if no better source is found.) --Kompik (talk) 12:25, 20 March 2008 (UTC)

Could someone else please check me on this one - It looks like the inclusions at the top of the article are running the wrong way. The integers form a PID, for example, but they certainly don't contain the field of rationals, which is what the chain seems to imply.Ike Benjamin (talk) 16:19, 21 September 2008 (UTC)

Hello Ike Benjamin, the inclusions are correct. In your example, what they indicate is that the class of PIDs contains the class of fields, that is, every field is PID - they do not mean that every PID contains a field or so. Ninte (talk) 09:47, 6 February 2009 (UTC)

Changes Needed

This article lacks many links to other articles, for example what does $\inf _{x\in R\setminus \{0\}}$ mean? I know that it's the infimum, but would a casual reader? If the article isn't meant for casual readers - which of course it is - then the slow and informal introduction is not needed.

There is also a clash of notation. The symbol $\mathbb {Z}$ is used for the integers at the start of the article, but Z is used later. Again, the notation $\mathbb {Z} ^{\geq 0}$ must be explained for the benefit of the casual reader.

 Δεκλαν Δαφισ   (talk)  17:51, 18 June 2009 (UTC)

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)

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)

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)

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)