From: Dave Rusin
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