From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3207 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: fundamental theorem of algebra Date: Sat, 1 Apr 2006 10:44:05 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019157 7706 80.91.229.2 (29 Apr 2009 15:32:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:37 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Sat Apr 1 19:19:08 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 01 Apr 2006 19:19:08 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FPpFN-0001cM-Q4 for categories-list@mta.ca; Sat, 01 Apr 2006 19:11:09 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 3 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:3207 Archived-At: On Thu, 30 Mar 2006, Vaughan Pratt wrote: > Regarding 3, the authors of the Britannica article seemed not to think > so, but perhaps this just reflects Garrett Birkhoff's attitude that "I > don't consider this algebra, but this doesn't mean that algebraists > can't use it" cited by Michael Artin when proving FTAlg in his 1991 book > "Algebra". Who on this list considers the fundamental theorem of > algebra "not algebra"? > The theorem is algebra, but its proof isn't: any proof has to involve some topological input (though that can be reduced to the Intermediate Value Theorem). Vaughan seems to have a problem with the phrase "elementary algebraic proof": of course, not all elementary proofs are algebraic (and not all algebraic proofs are elementary), and it is the word "algebraic" that matters here. Incidentally, I used that Birkhoff quote in the Introduction to "Stone Spaces" (1982). Did Mike Artin get it from me, or did he discover it independently? Even more incidentally, the first published proof of the Fundamental Theorem is not by Gauss. It appears in the only mathematical paper (in Phil. Trans. Roy. Soc. volume 88, 1798) of the Reverend James Wood, who was then a Fellow (and subsequently Master) of St John's College, Cambridge. (His other publications were all theological -- he was a Doctor of Divinity.) Wood's argument is essentially the same as Gauss's second proof (1816); by modern standards, what he writes in the paper doesn't constitute a rigorous proof, but (to quote the late Frank Smithies) "anyone reading Wood's paper must end up with the conviction that there is a proof somewhere there". Peter Johnstone