[FOM] Only one proof

[joeshipman@aol.com]

The F T of Algebra as a theorem of pure algebra (Gauss's 2nd proof, 
1816) states that in any field of characteristic zero in which all odd 
degree polynomials have roots and every element is a square or the 
negative of a square, all polynomials factor into linear and quadratic 
factors (and thus the field, if not algebraically closed, becomes 
algebraically closed when a square root of -1 is adjoined, by the 
quadratic formula). To apply this to show that the complex numbers are 
algebraically closed is almost trivial, and the fact that real 
polynomials of odd degree have a real root does not really require 
topology (it can be proved directly using the definition of reals using 
Dedekind cuts, so it only requires topology to the extent that topology 
is already required to define the real numbers). This is so far from 
some other proofs of the F T of Algebra (for example deriving it as a 
corollary of Liouville's theorem that a bounded entire function is a 
constant and so the reciprocal of a nonconstant polynomial cannot be 
entire) that it is silly to say there is essentially only one proof.

In 2006 I improved Gauss's proof by showing that "odd degree" could be 
replaced with "odd prime degree" and the restriction to characteristic 
0 could be removed (so any field whatsoever in which all prime degree 
polynomials have roots is algebraically closed). (See my article in the 
Fall 2007 Mathematical Intelligencer.)

The F T of Arithmetic has essentially different proofs according to 
Conway's "scope test": two proofs P1 and P2 are different if their most 
natural and obvious generalizations P1' and P2' [that is, 
generalizations requiring no new ideas] have different domains of 
applicability. I am currently working on the details of this analysis 
for that theorem; Conway and I recently completed a paper that analyzes 
7 essentially distinct proofs of the irrationality of the square root 
of 2.

-- JS

-----Original Message-----
From: Vaughan Pratt <pratt at cs.stanford.edu>
As we know from Gauss there are strikingly different proofs of the
Fundamental Theorem of Algebra (that every univariate polynomial with
complex coefficients has a complex root), although they all rely on
topology.
...
But what about the Fundamental Theorem of Arithmetic?  All proofs I'm
aware of seem to boil down to one version or another of Euclid's GCD
algorithm.  Is there a different proof?