From: Helmut Kahovec <helmut.kahovec@teleweb.at>
Subject: Re: Cyclotomic polynomial x^p-1, with p a Fermat prime
Date: Fri, 07 Jul 2000 00:31:14 GMT
Newsgroups: sci.math
Summary: [missing]

Gerry Myerson wrote:

>| In article <7jh5ls4pcoap06avobfl70mcjvufn4qc1t@4ax.com>, Foghorn
>| Leghorn <foghorn@netcene.com> wrote:
>| 
>| > The web page
>| >       http://library.wolfram.com/examples/quintic/timeline.html
>| > mentions that the cyclotomic polynomials x^17-1, x^257-1, and
>| > x^65537-1 have been solved with radicals. Can anyone point me to an
>| > explanation of these solutions (especially an on-line one)?
>| 
>| Let's consider x^17 - 1, as the others are similar. What's going on
>| is that are fields E, F, K, and L defined as follows:
>| 
>| There is a rational number d such that E is all the numbers of the
>| form a + b sqrt(d), with a and b rational;
>| 
>| There is a number e in E such that F is all the numbers of the form
>| a + b sqrt(e), with a and b in E;
>| 
>| There is a number f in F such that K is all the numbers of the form
>| a + b sqrt(f), with a and b in F;
>| 
>| There is a number k in K such that L is all the numbers of the form
>| a + b sqrt(k), with a and b in L.
>| 
>| Thus, every element of L can be expressed in radicals, in fact, all
>| you need is square roots (of square roots (of square roots (of square
>| roots)))). Among the numbers in L are all 17 solutions of x^17 - 1.
>| Thus, x^17 - 1 can be solved in radicals.
>| 
>| For x^257 - 1 & x^65537 - 1 you just have to go a few more steps.


Hello Gerry,

Well, contrary to the web page quoted above, John Stillwell writes in
his book "Elements of Algebra" (cf. [1], p.161, last paragraph):

"Deciding whether 65537 is the last Fermat prime may well tax the best
mathematicians of the future. In the meantime, there is a smaller
problem about 65537 which has also not been solved: express zeta[65537]
in terms of square roots! Expressions for zeta[17] and zeta[257] were
worked out early in the 19th century, and a Professor Hermes of
Göttingen is said to have spend 10 years in an unsuccessful attempt to
compute zeta[65537]. Apparently the computations became too big for him.
Surely today, with computer algebra systems, this should not longer be a
problem. If 65537 is indeed the last Fermat prime, whoever computes
zeta[65537] should earn at least a footnote in the history of
mathematics."


With kind regards

Helmut

Reference:

[1] John Stillwell, "Elements of Algebra", 2nd corrected printing,
Springer Verlag 1996, ISBN 0-387-94290-4