From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3214 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: fundamental theorem of algebra Date: Sun, 02 Apr 2006 14:43:09 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019161 7725 80.91.229.2 (29 Apr 2009 15:32:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:41 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Sun Apr 2 20:44:55 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 02 Apr 2006 20:44:55 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FQCEw-0003so-R8 for categories-list@mta.ca; Sun, 02 Apr 2006 20:44:14 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 10 Original-Lines: 69 Xref: news.gmane.org gmane.science.mathematics.categories:3214 Archived-At: For the bookworms among the readers of this FToA thread, let me offer four older references to undergraduate-accessible expositions of proofs along the lines already mentioned: First, in Birkhoff & Mac Lane (my own undergraduate algebra text), Section 3 of Chapter V of the 1953 ("revised") edition offers a proof along winding number lines on pp. 107-109. Next, in the 1975 MIR English edition of Kurosh's Higher Algebra (described as the "second printing"), section 23 of Chapter 5 offers a proof relying on the D'Alembert Lemma (on pp. 142-151). In the same Kurosh volume, moreover, section 55 of Chapter 11 offers a proof along symmetric function lines on pp. 337-340. Finally, one may find the Artinian proof in the real-closed fields section of van der Waerden's pre-WWII classic, Modern[e] Algebra. I refrain from citing other textbooks, and I remark that numberings (of pages, sections, chapters) may differ in other editions. Cheers, -- Fred Prof. Peter Johnstone wrote: > 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 > > >