Example 1 Let $z$ be a 7th root of 1, $z \neq 1$, find the value of $z+z^{2}+z^{4}$.


Solution: Let $u=z+z^{2}+z^{4}, v=z^{3}+z^{5}+z^{6}$, by $z^{7}=1, z \neq 1$ we have
$$
1+z+z^{2}+z^{3}+z^{4}+z^{5}+z^{5}=\frac{1-z^{7}}{1-z}=0,
$$

thus $u+v=-1$. Also, $u v=\left(z+z^{2}+z^{4}\right)\left(z^{3}+z^{5}+z^{6}\right)=3$, so $u, v$ are the roots of the equation $x^{2}+x+3=0$, hence
$u=\frac{-1 \pm \sqrt{7} i}{2}$, which means the value of $z+z^{2}+z^{4}$ is $\frac{-1 \pm \sqrt{7} i}{2}$.