Find all solutions for $z^3 = \overline{z}$
I know that $z = a + ib$ and that $\overline{z} = a - ib$, but when I try and calculate the solutions I get an unsolvable equation.
Would appreciate any help.
Forget about real and imaginary parts and note that every solution $z$ is such that $|z|^3=|z^3|=|\bar z|=|z|$ hence $|z|=0$ or $|z|=1$. Furthermore, $z^4=z^3\cdot z=\bar z\cdot z=|z|^2$.
Can you finish in both cases?