making conclusions from an equation in complex numbers.

Question

let $z \in C$ such that $z^3 = \overline{\rm z}^3$. then we can say that:

A. $\Im(z) = 0$ .

B. $\Re(z) = 0$ .

C. $|z| = 1$ .

D. None of the above.

I was able to disprove A,C. how can I disprove B? (the answer is D according to my school book)

Answer 1

Try a counter example: $z = 1$

$z^3 = 1$ and $\bar{z} = 1 \rightarrow \bar{z}^3 = 1$.

So $z^3 = \bar{z}^3$ but $Re(z) = 1$

Answer 2

WLOG $z=re^{it}$ where $r\ge0,t$ are real

We need $r^3e^{3it}=r^3e^{-3it}$

If $r\ne0,$

$$e^{6it}=1=e^{2n\pi i}\implies t=\dfrac{n\pi}3,0\le t<6$$