From: Dave Rusin <rusin@math.niu.edu>
Date: Thu, 29 Oct 1998 09:54:50 -0600 (CST)
To: brandsma@twi.tudelft.nl
Subject: Re: algebraic numbers query
Newsgroups: sci.math

In article <363868DF.3A7C99A5@twi.tudelft.nl> you write:

>Is every number of the form sin(r*Pi), where r is a rational number,
>algebraic?

If r=p/q this is  (z^p-zbar^p)/(2i)  where  z =exp(2 Pi i/q)  is a 
primitive q-th root of unity. In particular, yes, this is an algebraic
number (even an algebaric _integer_ after doubling).

Note that all such numbers with  p/q  in lowest terms are conjugate over  Q
(that is, they're all  (zeta - 1/zeta)/(2i)  for one of the several
conjugate primitive q-th roots of unity  zeta); in particular, they all
have the same minimal polynomial.

>If so, is there a general formula for its minimal polynomial over Q?

Probably; I don't know it. (You'll have to accept the cyclotomic polynomials
as part of the answer, surely -- do you accept those as part of a
"general formula"?).

One answer:  Prod(X - x^sigma), the product taken over the Galois group.
This is  Prod(X - sin((p/q)*Pi)), the product taken over  p  in  (Z/qZ)^\times.

There's probably a trick similar to the one for cosines: take the minimal
polynomial of  zeta,  which is the cyclotomic polynomial; divide by the
_square root_ ("half") of the highest power of  X  --  by symmetry you'll
get a sum of multiples of  (X^k + 1/X^k).  Expand as polynomials in  X+1/X,
and you have your minimal polynomial of  zeta+1/zeta.

dave