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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 BĂ©zout's identity are easily computable as soon as one has an algorithm for
Euclidean
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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. Knowledge is an
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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
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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.
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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.
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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...?
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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
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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
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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
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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
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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.
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So, I'll revert again the recent addition, and wait for a consensus for removing the definition of multiplicative
Euclidean functions.
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that seems to not have been mentioned in any textbook. So, unless reliable secondary source are provided, this is
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instantiating Bezout's identity. It would be nice if the language of the paragraph made this more explicit.
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Please, place the new sections at the end of the talk page and sign your posts with four tildes (~~~~).
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So, I suggest to remove this paragraph unless a source is provided for the notability of the concept.
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