8. Let $z$ be a complex number, $z^{7}=1$. Then $z+z^{2}+z^{4}=$
8. $\frac{-1 \pm \sqrt{7} i}{2}$.
Let $A=z+z^{2}+z^{4}$.
From $1+z+z^{2}+\cdots+z^{6}=\frac{1-z^{7}}{1-z}=0$, we can assume the dual quantity $B=z^{3}+z^{5}+z^{6}$.
It is calculated that $A+B=-1, A B=2$.
Thus, $A$ and $B$ are the roots of $x^{2}+x+2=0$.
Solving, we get $A=\frac{-1 \pm \sqrt{7} \mathrm{i}}{2}$.