Question

find all complex numbers $z$ which satisfy the following equation
Find all non-zero complex numbers $z$ satisfying $\overline{z}={iz}^2$.

Answer 1

$\overline{z}=i{z}^2$

$\Rightarrow |\overline{z}|=|i{z}^2|\Rightarrow |z|=|z{|}^2\Rightarrow |z|=1$

$\overline{z}=i{z}^2\Rightarrow z|\overline{z}|=i{z}^3$

$i{z}^3=1\Rightarrow z^3=-i=i^3$

$\Rightarrow z=i,iw,i{w}^2$ where $w\ne 1$ is a cube root of unity.

$\Rightarrow z=i,\frac{\sqrt{3}}{2}-\frac{i}{2},-\frac{\sqrt{3}}{2}-\frac{i}{2}$