Talk:Euclidean domain

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Changes Needed[edit]

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)[reply]

Small Confusion[edit]

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[edit]

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[edit]

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[edit]

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]