From: utzoo!decvax!ucbvax!G:shallit
Newsgroups: net.math
Title: Re: Questions about pizza answer
Article-I.D.: populi.215
Posted: Wed Jun 23 10:37:06 1982
Received: Sun Jun 27 04:42:14 1982

[1]  You hit the nail on the head.  You CAN'T construct the 9-th roots,
since there is no way to get them from cube roots.  The only roots
you can construct are products of a power of 2 and a set of Fermat
primes.  That set includes at most 1 of each prime.  In particular,
you can't construct the roots corresponding to powers of Fermat
primes.

[2]  With respect to actually finding the roots of the polynomials in
question, this can be done.  The procedure is not terrible easy to
describe, but I can give a reference:  Jacobson's Basic Algebra I,
pp. 215-218.  I must admit this is rough going.  But really one needs
to be conversant with Galois theory to REALLY understand what is
going on here.  Herstein's Topics in Algebra is a good intro to
Galois theory.
/Jeff Shallit, Department of Mathematics, University of California, Berkeley