NetNews Usenet Archive 1993 #1/NN_1993_1.iso/spool/sci/math/17943

Text File | 1993-01-10 | 3.3 KB | 67 lines

Newsgroups: sci.math
Path: sparky!uunet!paladin.american.edu!darwin.sura.net!jhunix.hcf.jhu.edu!fmsrl7!destroyer!ncar!isis.cgd.ucar.edu!steele
From: [e-mail] (Alfred Steele)
Subject: Euclidean domains
Message-ID: <[e-mail]>
Sender: [e-mail] (USENET Maintenance)
Organization: Climate and Global Dynamics Division/NCAR, Boulder, CO
Date: Mon, 11 Jan 1993 00:53:20 GMT
Lines: 56


In article <[e-mail]> you write:
|>>> |>Why there are not other candidates for the Euclidean function?
|>
|>I'm not sure what the original question was, but hopefully the following
|>is not altogether irrelevant!
|>
|>A Euclidean domain is a ring equipped with a special kind of a valuation -- an
|>euclidean valuation, essentially one that makes it amenable to carry out the
|>Euclidean algorithm. As usual, one can get a norm out of this by taking a nega
tive
|>exponential. Ostrowski's classic theorem asserts that the only possible norms
on
|>the rationals are the absolute value and the p-adic norms obtained from p-adic
|>valuations. I am unaware of of similar general normal form results about
|>euclidean valuations on various classes of rings. Perhaps a book on Valuation
|>Theory (such as Endler's, which unfortunately is inaccessible to me at present
)
|>might shed some light on this.


Functions for Euclidean rings satisfy an inequality.  The multiplicative
property of norms is essential for something like Ostrowski.  The post
by R. T. Bumby shows us that we are on the wrong track.  I will repeat it.
The problem in Hungerford would be easy if one restricted oneself to
absolute value norm but if you take it at face value and allow the function
to be arbitary (except for the conditions) it might be very difficult.
Hay - everything is Hungerford is not correct - he has other errors.

If I wanted to really find out about the answer I would get off my butt an
go check out the references in the following post:

|>This should be a FAQ.  T. S. Motzkin, "The Euclidean Algorithm", Bull.
|>Amer. Math. Soc. 55(1949), 1142-1146, gave a simple analysis of the
|>properties of any Euclidean Algorithm in an integral domain.  The idea
|>is to work backwards, starting with the set consisting only of zero,
|>and applying the following construction.  The derived set of a set, S,
|>consists of all elements of the domain which have a complete set of
|>residues in S.  This construction may be extended transfinitely if
|>necessary by taking unions at limit ordinals.  In order to have a
|>Euclidean Algorithm, you must be able to exhaust the domain in this
|>way.  For quadratic number rings, there are only finitely many units.
|>The derived set of {0} consists only of units.  If all proper ideals
|>have index greater than the number of units (only 2 except for some
|>rings that are already Euclidean for the norm), the process stops
|>there.
|>
|>Another major article on Euclidean Algorithms is P. Samuel, "About
|>Euclidean Rings", J. Algebra 19 (1971), 282-301.  It would appear that
|>the next major exposition is due this year.
|>--
|>R. T. Bumby **  Rutgers Math ||   Amer. Math. Monthly Problems Editor
|>[e-mail]       || P.O. Box 10971 New Brunswick, NJ08906-0971
|>[e-mail]     || Phone: [USA] 908 932 0277 * FAX [phone]

Alfred T. Steele ([e-mail])