From: utzoo!decvax!harpo!floyd!rjs
Newsgroups: net.math
Title: Questions about pizza answer
Article-I.D.: floyd.306
Posted: Wed Jun 23 09:30:54 1982
Received: Sun Jun 27 03:47:57 1982

The answer to the pizza question was instructive (and in fact showed
that you cannot divide a circle into 7 equal parts with a straight
edge and compass).  However, it does not show how to accomplish
divisions it claims are possible.  My questions are:

I can see how to construct the roots of x^pq - 1 = 0 from the
roots of x^p - 1 = 0 and x^q - 1 = 0 if p and q are mutually
prime (all products of roots, one from each set are roots).
But what if they are not mutually prime?  Then there are
roots (other than 1) in common and multiplying roots doesn't
generate enough roots.  I.e. how do you get the roots of
x^9 - 1 = 0 from the roots of x^3 - 1 = 0?

Once you try to get the roots for a Fermat prime > 3 you have
the problem of finding roots of a polynomial of degree > 2.
This is usually quite difficult, but all polynomials of interest
are of the form 1 + x + x^2 + x^3 + ... + x^2^2^n.  Is there an
easy (or even known) method for finding these roots?

Thanks.

Robert Snyder, III   (harpo!floyd!rjs)