Dan Asimov: I learned (or was reminded) only recently that the real and imaginary parts of any nth root of unity can each be expressed in terms of (an iterated rational expression in) radicals of integers (using only roots lower than the nth). For instance, Wikipedia lists the real part of (exp(2pi*i/7)) as follows: cos(2pi/7) = (-1 + ((7 + 21*sqrt(-3)/2)^(1/3) + ((7 - 21*sqrt(-3)/2)^(1/3))/6 --WDS: Gauss famously showed that the nth roots of unity are constructible iff the odd part of n is a product of distinct Fermat primes. The truth of Asimov's claim follows IF we know that the cyclotomic polynomials always have solvable Galois group. I believe the Galois group for them always arises from (Z mod m), hence is abelian, hence solvable, thus proving Asimov's claim. https://en.wikipedia.org/wiki/Cyclotomic_polynomial http://www.math.niu.edu/~beachy/aaol/galois.html#cyclotomic https://en.wikipedia.org/wiki/Solvable_group -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)