Alternatively from z^10 - 1 = 0, a primitive 10th root of unity must satisfy x^5 + 1 = 0 (otherwise, it's not primitive). Noting that x = -1 is a root, we find that the four primitive fourth roots of unity satisfy (x^5 + 1)/(x + 1) = x^4 - x^3 + x^2 - x + 1 = 0. The product of the roots equals 1 (from the constant term), and the sum of the roots equals 1 (being -1 times the coefficient of x^3).
Alternatively from z^10 - 1 = 0, a primitive 10th root of unity must satisfy x^5 + 1 = 0 (otherwise, it's not primitive). Noting that x = -1 is a root, we find that the four primitive fourth roots of unity satisfy (x^5 + 1)/(x + 1) = x^4 - x^3 + x^2 - x + 1 = 0. The product of the roots equals 1 (from the constant term), and the sum of the roots equals 1 (being -1 times the coefficient of x^3).