Re: fundamental theorem of algebra

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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