Determine the generator for the cyclic group formed by the solutions of $x^9 = 1$. 
Find the solutions to $x^9 = 1$ and determine the generator for the cyclic group formed by the solutions.

The equation can be factored as $(x^3 - 1)(x^6 + x^3 + 1) = 0$ and the solutions are $$\begin{align*}
    x &= 1\\
    x &= -\sqrt[\leftroot{1}\uproot{3}9]{-1} \\
    x &= (-1)^{2/9} \\
    x &= -\sqrt[\leftroot{1}\uproot{3}3]{-1} \\
    x &= (-1)^{4/9} \\
    x &= -(-1)^{5/9} \\
    x &= (-1)^{2/3} \\
    x &= -(-1)^{7/9} \\
    x &= (-1)^{8/9}
\end{align*}$$ but I'm unsure how to determine the generator. How are these found generally and what would be my options to start determining it?

Note that
$x^9=1=\cos 2n\pi +\sin 2n\pi=e^{i2n\pi}$
$\implies x=e^{i2n\pi/9}; n=0,1,2,...,8$ are  roots.
Assume $e^{i2\pi/9}=a$. Then the powers of it coprime to $9$ viz. $a,a^2,a^4,a^5,a^7 $ and $a^8$ are generators.
