Re: WHY ARE WE CONCERNED? I

[Vaughan Pratt <pratt@cs.stanford.edu>]

In response to Peter Johnstone (and those who responded privately), my
point about the Fundamental Theorem of Algebra was not that this
particular proof (based on the limiting behaviors of small and large
circles) was not known to anyone, but that it had not emerged, instead
being effectively sat on by those in the know, even if not
intentionally.  At this risk of sounding like an Abu Ghraib
interrogator, "who knew?"

My claim is that no extant proof at all, that or any other, was
considered fit for an elementary exposition more than a couple of
decades ago.  If that estimate is right, the 1982 Pontrjagin article
cited by Nikita Danilov would be one of the earliest popular expositions
based on the circles argument, assuming the section containing Fig. 6 is
the relevant one (my Russian is even rustier than my algebra).  I'd be
very interested in seeing an earlier popular account that didn't claim
that every proof necessarily either was long or depended on out-of-scope
material.

As a case in point, just now I checked a relatively recent Brittanica
article on algebra (1987 ed.), which states flatly (p.260a) that "No
elementary algebraic proof of [the FTAlg] exists, and the result is not
proved here."  (Not even "is known" but "exists"; an expository article
should not assume that the reader knows the jargon meaning of this term
as "exists in the literature".)  The authors taking responsibility for
this claim were Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter
Hilton, and Paul Cohn.  They go into detail to show that z^n = a has n
roots, starting with the geometry of addition and multiplication in the
Argand diagram, so it's not as if their exposition was at too elementary
a level to talk in terms of mapping circles, or that "algebraic" ruled
out simple geometric arguments.

I submit their nonexistence claim as prima facie evidence for my claim
that the very few who knew this argument weren't even letting the likes
of Birkhoff, Hall, etc. in on it, let alone "the rest of us."

The general message in the literature prior to the 1980's seemed to be,
if Gauss couldn't find a simple proof in half a dozen tries, there isn't
one.  If you don't possess the necessary higher maths or the stamina for
an intricate argument, we can't help you with that result, ask us about
solvability of z^n = a.

Good for Pontrjagin for promoting FTAlg to school children!

Vaughan Pratt