We have x3 + 1 = (x + 1) (x2 - x + 1) . Therefore, a and b are the complex cube roots of –1 so that we may take α = –ω and β = –ω2, where w ≠ 1 is a cube root of unity. Thus α100 = (−ω)100 = ω and β100 = (−ω2)100 = ω2 , so that the required equation is x2 + x + 1 = 0