From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Algebraic Numbers
Date: 14 May 1997 20:56:33 GMT

In article <5knvic$nrt@uni00nw.unity.ncsu.edu>,
Gregory Alan Gibson <gibsong@aurora.ncat.edu> wrote:

>    My question is which numbers on the unit circle in the complex plane
> are algebraic?  Since e^(xi) = cos(x) + i(sin(x)), I'm curious as to 
> which values of x make e^(xi) an algebraic number?  (0 <= x <= 2*Pi)
>
> Clearly rational multiples of Pi work since e^(q*Pi*i) = (e^(Pi*i))^q =
> (-1)^q, which is algebraic if q is rational.  Are there any other values
> for x which produce an algebraic value for e^(xi)?  

If  z  is any complex number which is algebraic over the rationals, then
its complex conjugate  zbar  is also algebraic, and then their product
R=z*zbar  and its square root  r=sqrt(R)  are also algebraic. Thus
z/r  is algebraic, and clearly on the unit circle.

Moreover, every algebraic number on the unit circle is clearly obtained
by this process, since if  |z|=1  in the first place then  z/r=z.

dave