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.

Jelly, 14 bytes

Rg=1O°÷×ı360Æe

Try it online!

How it works

z = e^2tπi is an n^th root of 1 if and only if t = k / n for some integer k.

z is primitive if and only if k and n are coprime.

Rg=1O°÷×ı360Æe  Main link. Input: n

R               Yield [1, ..., n].
 g              Compute the GCDs of reach integer and n.
  =1            Compare the GCDs with 1.
    O           Get all indices of 1's.
                This computes all the list of all k in [1, ..., n] 
                such that k and n are coprime.
     °          Convert the integers to radians.
      ÷         Divide the results by n.
       ×ı360    Multiply the quotient by the imaginary number 360i.
            Æe  Map exp over the results.

Julia, 48 bytes

n->cis(360deg2rad(filter(k->gcd(k,n)<2,1:n))/n)

This is a lambda function that accepts an integer and returns an array of complex floats. To call it, assign it to a variable. It uses the same approach as Dennis' Jelly answer.

Ungolfed:

function f(n::Int)
    # Get the set of all k < n : gcd(k,n) = 1
    K = filter(k -> gcd(k,n) < 2, 1:n)

    # Convert these to radian measures
    θ = deg2rad(K)

    # Multiply by 360, divide by n
    θ = 360 * θ / n

    # Compute e^iz for all elements z of θ
    return cis(θ)
end