Euclidean domains

January 23, 2011

We all learn in intro abstract algebra that a euclidean domain is a PID. It turns out that the converse is almost true. Namely, if one relaxes the definition of a euclidean norm (instead of a euclidean algorithm, you have something a bit weaker) you get something entirely equivalent to being a PID. This is apparently due to Greene in the Monthly, 1997 (and has a quick proof). Now, this material is in ch. 1.

Lang’s Algebra (as well as some of his other books, too) has a lot of these kinds of isolated references to scattered results in the literature. Some of these are quite interesting; it is probably worth adding more of these. Doing so will also make the book less “canonical”!

It happens, coincidentally, that we also got a donation on euclidean domains, which has been partially merged in.


2 Responses to “Euclidean domains”

  1. The fact that being a PID is equivalent to the existence of a certain kind of norm which is reminiscent of a Euclidean norm but slightly weaker goes all the way back to Dedekind and was independently rediscovered by Hasse. See Section 8.3 of my survey article on factorization for more information: (I don’t know why Greene’s article was published without any reference to this not-especially-obscure older work. It looks like an oversight to me…)

    • Akhil Mathew Says:

      Dear Pete, thanks for pointing that out! I’ll fix it when I get some time (which reminds me that I have not worked much on this in the past week or so owing to other commitments — i.e. large loads of homework).

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