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