Show $R^*$ is not a cyclic group 
$R = \mathbb{F}_3[x]/\langle X^3+X^2+1\rangle$ and $\alpha=[X]$ in $R$. How do you prove that the group $R^*$ is not cyclic?

We have shown that $\alpha$ is a unit in $R$ with order $8$ and that $\alpha^4$ and $-\alpha^4$ are two different elements in $R^*$ both with order 2.

Hint:
$x^3+x^2+1$ factors into the irreducibles $(x-1)$ and $(x^2+2x+2)$ over $\Bbb F_3$.
By the Chinese Remainder Theorem, $R\cong \frac{\Bbb F_3[x]}{(x-1)}\times \frac{\Bbb F_3[x]}{(x^2+2x+2)}$.
Can you see what the two pieces look like, and how you can use this to see the units of $R$?
