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 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