Wikipedia:Featured article candidates/Euclidean algorithm/archive1: Difference between revisions

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::I had a vague feeling Fowler had a point there, so I dug through some books I had handy. I found that (as expected) Hardy & Wright's number theory book states that "Euclid's algorithm" is defined by generating the sequence of remainders which terminates (it seems to give no name for the "division algorithm", merely calling it division with remainder at times), Dummit & Foote's abstract algebra book states that the "Euclidean algorithm" comes from the ''division algorithm'', but Herstein's ''Topics in Algebra'' does indeed call the above, ''the'' Euclidean algorithm. Herstein is a pretty well-known algebra book, so I dug a bit more and I found that a source on the [[Euclidean ___domain]] article, which was published in the Bulletin of the American Mathematical Society in 1949, also uses the term Euclidean algorithm for division algorithm (see [http://projecteuclid.org/euclid.bams/1183514381]). I think in number theory books there is a pretty well-established tradition of using "Euclidean algorithm" to mean generating the sequence of remainders from the two initial numbers. In algebra texts that discuss Euclidean domains, my suspicion is that books here and there may use Euclidean algorithm to mean division algorithm but I suspect that modern books generally don't; I find Dummit & Foote is pretty reliable regarding modern terminology, while Herstein is from 1961 and sometimes a bit outdates on terminology. In any case, I think a note or footnote is in order. --[[User:C S|C S]] ([[User talk:C S|talk]]) 04:59, 29 April 2009 (UTC)

:::Yes, I guess you are right: number theory texts do call it the "division algorithm," and in fact, come to think of it, in high-school we called it the "division algorithm" ourselves, perhaps because it was introduced as a part of elementary number theory. Somewhere in college though the name became less certain (as I remember it). As for texts, among the classic algebra texts, both van der Waerden and Birkoff/Mac Lane do call it the "division algorithm," but Herstein doesn't (as you say). The more recent texts seem to be a mixed bag. I don't know Dummit and Foote, but among the books published in the last 20 years that refer to the "division algorithm" as the "Euclidean algorithm" are (the links should take you straight to the page about the "Euclidean Algorithm"): Hilton and Wu's ''[http://books.google.com/books?id=ua5gKZt3R6AC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA10,M1 A Course in Modern Algebra]'' (1989), Rowen's [http://books.google.com/books?id=EmO9ejuMHNUC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA116,M1 Algebra: Groups, Rings, and Fields] (1994), Lang's ''[http://books.google.com/books?id=Fge-BwqhqIYC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA173,M1 Algebra: A Graduate Course]'' (2002), Murty and Esmonde's [http://books.google.com/books?id=YaqVpdrngNYC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA3,M1 Problems in algebraic number theory] (2004), Lang's ''[http://books.google.com/books?id=PdBirmNTwu0C&printsec=frontcover&source=gbs_summary_r&cad=0#PPA113,M1 Undergraduate Algebra]'' (2005), and Lowen's [http://books.google.com/books?id=8svFC09gGeMC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA27,M1 Graduate Algebra: The Noncommutative View] (2008). Lang (2002), in particular, is still widely used, I think. So the footnote will be useful for any others who have questions like mine. :) Thanks! [[User:Fowler&amp;fowler|<font color="#B8860B">Fowler&amp;fowler</font>]][[User talk:Fowler&amp;fowler|<font color="#708090">«Talk»</font>]] 12:26, 29 April 2009 (UTC)

;!Votes