re: fundamental theorem of algebra

["John Baez" <baez@math.ucr.edu>]

Dear Vaughan -

You write:

> 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."

I really doubt those authors were unaware of the topological proof
of the fundamental theorem of calculus in 1987.  After all, it's
exercise H.5 in chapter 1 of Spanier's "Algebraic Topology", copyright
1966.  This book used to be the canonical textbook on algebraic
topology, and Peter Hilton is a darn good algebraic topologist.

I think I learned this topological proof sometime in grad school,
around 1986.  So, I don't think it was any sort of secret by then.

I don't know what counts as an "elementary algebraic proof", but
people often say that there is no "purely algebraic proof" of the
fundamental theorem of calculus.   After all, this theorem is about
the complex numbers, which are often defined in terms of the real
numbers, which are often defined as a topological completion of the
rational numbers.  I hope this is what the Encyclopedia article
was trying to say.

There are some so-called "algebraic proofs":

http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

that use a bare minimum of topology.  These proofs tend to have
a purely algebraic core, namely "if odd-degree polynomials
and the polynomial x^2 + 1 have roots in some field, this
field is algebraically closed".  But, they use the intermediate
value theorem for continuous functions f: [0,1] -> R to show
that C meets these conditions.  So, I wouldn't call them "purely
algebraic".

It's sort of ironic that the so-called "fundamental theorem of
algebra" doesn't have a purely algebraic proof.

Gauss is famous for having given a proof of the fundamental
theorem of algebra in his dissertation back in 1799.  On the
St. Andrews math history website they write:

  Gauss's proof of 1799 is topological in nature and has some
  rather serious gaps.  It does not meet our present day
  standards required for a rigorous proof.

They don't say how the proof went.  So, I decided to find out!
I was hoping I could irritate you by showing that it was just
the topological proof you claim is so new.  There's a discussion
of it here:

Hans Willi Siegberg
Some Historical Remarks Concerning Degree Theory,
American Mathematical Monthly, 88 (1981), 125-139.
(Available on JSTOR, or via Google Scholar.)

As the title hints, Gauss' proof uses ideas closely related
to the winding number.  Unfortunately, it's slightly different
than the proof you like.

The idea is to take a polynomial of degree n, say

P: C -> C

break it into real and imaginary parts

P = U + iV,

see where they vanish:

S = {z: U(z) = 0}
T = {z: V(z) = 0}

and show that the intersection of S and T is nonempty.

Gauss argues that far from the origin, S and T are smooth curves.
Because the leading term of the polynomial dominates the rest,
each of these curves intersects any sufficiently large circle
transversely at n points.

If we go around the circle these intersection points alternate:
first a point in S, then one in T, then one in S, and so on.

Moreover, the curves I'm talking about can't just disappear as we
follow them into the disk, since they separate the region where U
(resp. V) is positive from the region where it's negative.  They
may become singular, or intersect, but they can't just end!

"So", S and T must intersect somewhere.

This is true, but it takes more topology to prove it rigorously than
was available to Gauss.

Gauss knew his proof wasn't completely rigorous, so he invented some
other arguments.  The "winding number" idea you like is lurking in
Gauss' third proof, which he wrote up in 1816 - but he only gave this
winding number proof explicitly in 1840.  According to Siegberg,

  Indeed, in a lecture "Theorie der imaginaeren Groessen (1840),
  Gauss mentioned [see Fraenkel, 1922] that his third proof of
  the fundamental theorem of the algebra originated from his first
  one, and he gave the function-theoretic argument that the winding
  number W(P|S, 0) equals n [the degree of the polynomial], whereas
  the winding number of any map F: (B,S) -> (R^2, R^2 - {0}) vanishes
  if there is no zero of F in B [see Fraenkel, 1922].  However,
  this argument cannot be found explicitly in [Gauss, 1816].

So, I guess that except perhaps for Gauss, nobody knew the proof
you're talking about until 1840.

Best,
jb