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