[FOM] New Proof of Fundamental Theorem of Arithmetic

[joeshipman@aol.com]

I have two somewhat contradictory observations here:

1) There is plenty of buried treasure in well-plowed fields. I am often 
led to not only new proofs of old results, but proofs of new results, 
by reflecting deeply on very well-known mathematics. For example, I 
discovered that fields in which all polynomials of prime degree have 
roots are algebraically closed by trying to understand Gauss's original 
proof of the Fundamental Theorem of Algebra as well as I possibly 
could.

There were two key observations which allowed me to extend Gauss's 
proof (by eliminating the characteristic-0 hypothesis and the 
assumption that polynomials of odd composite degree have roots). The 
first was something I had observed when learning Galois Theory from 
Michael Artin in 1980 (namely, that although "most" nth-degree 
polynomials have the full symmetric Galois group of order n!, "most" 
algebraic numbers have a Galois group of the smallest possible order n, 
where the first sense of "most" is "density in coefficient space" and 
the second sense of "most" is "all the elements of an algebraic number 
field except for finitely many subspaces of lower dimension as vector 
spaces over Q). The second, which I learned from a (completely 
elementary) 1970 paper by Conway on "finite choice axioms", was that 
the way to relate symmetries acting universally on different 
cardinality sets was to look at the semigroup generated by the indexes 
of the proper subgroups of a finite group.

2) On the other hand, there is something meaningless about finding new 
proofs of results which follow infallibly from a general theory. 
Already in high school geometry I had learned that any theorem that 
could be stated in the standard language of Euclid could be proved by 
reducing everything to coordinates and applying analytic geometry, 
since all questions ultimately got reduced to questions about 
equalities and inequalities between real polynomials which were clearly 
answerable by standard calculus techniques. [This was of course 
formally established by Tarski in the 30's but long before then there 
had ceased to be any "open questions" in the subject because everyone 
implicitly understood this.]

The second point of view here has been taken to extremes by Doron 
Zeilberger, and I find it hard to gainsay him; yet the new proofs of 
results like Heron's formula somehow seem to improve our understanding. 
It is a puzzlement.

-- JS

-----Original Message-----
From: Vaughan Pratt <pratt at cs.stanford.edu>
(You'd think all the proofs of very old theorems had been found by now,
but I recently found a publishable proof of the interderivability of 
the
Pythagorean Theorem and Heron's area formula using Al-Karkhi's 11th
century quarter-squares rule, google for "Factoring Heron.")