From: nikl+sm000851@mathematik.tu-muenchen.de
Subject: Re: Algebraic integer on the unit circle
Date: 25 Nov 2000 12:57:24 GMT
Newsgroups: sci.math
Summary: [missing]

Yuval Dekel wrote:
>Suppose that z is on the unit circle of the complex plane and
>satisfies a monic equation with integer coefficients. Show that  z is
>a root of unity  i.e z^n = 1 for some n.

I won't... because it's not true.

For a somewhat famous counterexample, consider the 8 non-real
roots of Lehmer's polynomial, x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1,
and look up "Salem numbers" in the literature -- where you'll
also find proofs of Kronecker's theorem, according to which an
algebraic integer in C which *along with all of its conjugates*
lies on the unit circle must be a root of unity.  The idea is to
consider the set of all the powers of all the conjugates, and
observe that one has bounds for all the coefficients of the
minimal polynomials of the powers.

For a methodical way to construct counterexamples, let z be a
totally real, nonzero algebraic integer which has at least one
real embedding of abs. value < 2 and at least one of abs. value
> 2.  Then consider the roots of the equation X^2 - zX + 1.

Enjoy,
Gerhard