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: How many elements of order two have a cyclic group? If $R^*$ has two or more, is it compatible with it being cyclic?
