Re: [HM] Wantzel

Daniel E. Otero (otero@xavier.xu.edu)
Tue, 24 Nov 1998 15:58:39 -0500

> On Sun, 22 Nov 1998, Robin Hartshorne wrote:
>
> > Dear Colleagues
> >
> > Everyone knows that one of the successes of modern
> > algebra has been to prove the impossibility of the classical
> > problems of doubling the cube and trisecting the angle with
> > ruler and compass. The key point is that if a is
> > constructible, then a is a root of an irreducible polynomial
> > f(x) with rational coefficients and degree a power of 2.

It has occurred to me often in recent years that the resolution of the
classical problem of the quadrature of the circle came when Lindemann proved
the transcendence of pi in 1882. However, such a powerful is unnecessary: one
only needs to show that if K is a quadratic extension of the number field k,
and pi lies in K, then pi must lie in k. This, together with Lambert's result
(1761) on the irrationality of pi is enough! Is this *simpler* result really
not simpler?

Danny Otero
Xavier University
Cincinnati, OH

PS: Of course the answer to this question is YES!