From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: My FLT proof: Simple mechanics of it
Date: 12 Oct 2000 04:18:14 GMT
Newsgroups: sci.math

In article <8rsfak$heo$1@wisteria.csv.warwick.ac.uk>,
mareg@mimosa.csv.warwick.ac.uk () writes:
|This does not make any sense, because the equation x^p + y^p = z^p is
|false/meaningless in R[x, y]. The ring has to be
|
|R[x, y, z]/(x^p + y^p - z^p).
|
|I have no idea whether or not that is a UFD, even if R = integers.

One loses the UFD property rather easily this way. x^p+y^p=z^p
corresponds to a projective plane curve of genus > 0. The genus being
greater than 0 corresponds to certain limitations.

For example, z=0 cuts the curve in p points (cuts the surface in p
lines), most of them complex. We can cut the surface along one of them
with x+y=0. If this were a UFD, there'd be a common divisor of x+y and
z which was zero on (x, -x, 0) and nowhere else on the surface. Well,
having to divide z forces the "order" of vanishing, in a suitable
sense, to be just 1, and you can't get a polynomial which vanishes
just there and just to order 1.

So z^p = (x+y)(x^{p-1}-x*y^{p-2}+...+y^{p-1}) is a pair of
factorizations without a common refinement provided R satisfies some
mild conditions which I don't feel like working out right now.

Keith Ramsay