How to solve $z^{1995}+z^{-1995}$ if $1+z+z^2=1$?

Question

I know this is high school basic math, but please bear with me:

$z^2=-z$, so the $z_1=-1$ or $z_2=0$.

Now, when I put the solutions in $z^{1995}+z^{-1995}$ equation:
1. $z_1=-1 \implies (-1)^{1995}+(-1)^{-1995}=(-1)+(-1)=-2$.
2. $z_2=0 \implies 0^{1995}+0^{-1995}=0+$(not defined)$=0.$

But the solution is $z^{1995}+z^{-1995}$=2

What did I miss?

Answer 1

If $1+z+z^2=0$, then $z= e^{2\pi i/3}$, $e^{-2\pi i /3}$. This is because $z$ is a root of $z^3-1=0$ not equal to $1$. Note that $z^3=z^{-3}=1$ and $3|1995$.