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). 1) Is there an easy proof of this? 2) How do I get Mathematica to output these expressions? (Everything I've tried so far gets only the dumbest error messages.) 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 . I'm confident that several of the 2^2 3^2 = 36 choices of roots give the correct answer, but: which ones? This may be easy to guess, but how about cos(6pi/23) ? 3) There ought to be a notation that doesn't leave these choices unspecified. —Dan