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?

The multiplicative group of $9$-th roots is isomorphic to the additive group $\mathbf Z/9\mathbf Z$, under the isomorphism:
$$\mathbf Z/9\mathbf Z\longrightarrow U_9,\quad k\bmod9\longmapsto\mathrm e^{\tfrac{2ik\pi}9}.$$
Under this isomorphism, generators of $\mathbf Z/9\mathbf Z$ map onto generators of $U_9$, i.e. primitive $9$-th roots of unity. Therefore it is enough to know the generators of $\mathbf Z/9\mathbf Z$, which is standard.
