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