root of unity, prove relationship

Question

If $z^3+1=0$ and $\lambda$ is a complex root, prove that $\lambda^2 + 1 = \lambda$.

I have no idea how to start can someone please help me, I will be very grateful.

Thanks.

Answer 1

$z^3 + 1 = (z+1)(z^2-z+1)$ so the complex root of equation will satisfy the following $z^2-z+1=0$ $\implies z^2+1=z$ which is the required condition