Primitive Roots of Unity
Jelly, 11 9 bytes
Thanks to @Dennis for -2 bytes!
Rg=1O÷H-*

I wanted to generate the numbers coprime to N by folding set difference over all of the roots of unity from 1 to N, but I couldn't figure out how so I used @Dennis's method.
Rg=1O÷H-*         Monadic chain:          6
R                 Range                   [1,2,3,4,5,6]
 g                Hook gcds with range    [1,2,3,2,1,6]
  =1              [gcds equal to one]     [1,0,0,0,1,0]
    O             Replicate indices       [1,5]
     ÷H           Divide by half of N     [1/3,5/3]
       -          Numeric literal: - by itself is -1.
        *         Take -1 to those powers [cis π/3,cis 5π/3]

Try it here. Valid in this version of Jelly, but may not be in versions after February 1, 2016.