Consider the ring $R=\mathbb F_p[x]/(x^3-1)$. Find the group of units (as a product of cyclic groups) for $p=29,\ 31$.
We have $x^3-1=(x-1)(x^2+x+1)$. If I manage to prove that the ideals $(x-1)$ and $(x^2+x+1)$ are comaximal in $\mathbb F_p[x]$, then it will follow by the Chinese remainder theorem that $R\simeq\mathbb F_p[x]/(x-1)\times \mathbb F_p[x]/(x^2+x+1)$. I don't see how I can obtain $1$ as a linear combination of these generators, so the first question is how to show they are comaximal?
Secondly, $\mathbb F_p[x]/(x-1)\simeq \mathbb F_p$, and the answer depends on the second factor. It is a pain to check whether $x^2+x+1$ has a root for $p=29,\ 31$, so I was wondering how to find roots of the polynomials quickly or prove that they are irreducible?