1. Find all complex numbers $z$ such that
$$
P(x)=(x-z)\left(x-z^{2}\right)\left(x-z^{3}\right)
$$

has real coefficients.
1. Obviously, all real numbers satisfy the condition.

Below, assume $z$ is not a real number.
Since $P$ is a cubic polynomial, there exists a real root.
If $z^{\prime}$ is a root of $P$, then $z^{\prime}$ is also a root of $P$.
Thus, $\bar{z}=z^{2}$ or $\bar{z}=z^{3}$.
(1) From $|\bar{z}|=\left|z^{2}\right|(z \neq 0)$, we get $|z|=1$.
Then $z^{3}=z^{2} z=\bar{z} z=|z|^{2}=1$.
Thus, the factors of $P$ $(x-z)\left(x-z^{2}\right)$ and $x-z^{3}$ are all real coefficients.
From $z^{3}=1$ and $z \neq 1$, we get $z=\frac{-1 \pm \sqrt{3} \mathrm{i}}{2}$.
(2) From $|\bar{z}|=\left|z^{3}\right|$, similarly, we get $z^{4}=1$, thus, $z= \pm \mathrm{i}$.

Thus, the factors of $P$ $(x-z)\left(x-z^{3}\right)$ and $x-z^{2}$ are all real coefficients.

Therefore, all complex numbers $z \in\left\{\frac{-1 \pm \sqrt{3} \mathrm{i}}{2}, \pm \mathrm{i}\right\}$ and all real numbers.