Complex Numbers - Roots of Unity

The cube roots of unity are \(z_0=1\), \(z_1=-\tfrac12+\tfrac{\sqrt3}{2}i\) and \(z_2=-\tfrac12-\tfrac{\sqrt3}{2}i\). Their properties follow as below.

1. \(z_1\) and \(z_2\) are conjugates. So, \(z_2\) is a reflection of \(z_1\) in the real axis.
2. if \(\omega\) is a cube root of unity, then i.e. \(\omega^3 = 1\)
   1. \(\omega^m\) is also a cube root of unity, for any positive integral value of \(m\). \((\omega^m)^3 = \omega^{3m} = (\omega^3)^m = 1^m = 1\)
   2. If also \(\omega\ne1\), then \((\omega-1)(1+\omega+\omega^2)=\omega^3-1=0\), so \(1+\omega+\omega^2=0\). For \(\omega=1\), the sum is \(3\).
3. A calculator with a complex polynomial solver can solve \(z^3-1=0\). Verify the three roots algebraically; menu names depend on the model.
4. the cube roots of unity are represented by the vertices of an equilateral triangle inscribed in a unit circle with centre at the origin, with \(z_0\) on the real axis, and \(z_0\), \(z_1\) and \(z_2\) separated by equal angles of \(\frac{2\pi}{3}\).