What is the decomposition of the ring $\mathbb{F}_p[x]/(x^n-1)$? Let $p$ be a prime, and $n\ge 1$ an integer.
I'd like to decompose the ring $\mathbb{F}_p[x]/(x^n-1)$ into a direct product of artinian local rings.
I know we can write $x^n-1 = \prod_{d\mid n}\Phi_d(x)$, but how do the cyclotomic polynomials $\Phi_d(x)$ decompose mod $p$? I know their irreducible factors should have degree equal to the order of $p$ modulo $d$. Can $\Phi_d(x)$ have distinct irreducible factors (or do they always decompose as a power of an irreducible?)? Can $\Phi_d(x),\Phi_{d'}(x)$ share irreducible factors for $d\ne d'$?
Is there a nice way to write this decomposition?

A natural way is to look at your ring as the ring of $n\times n$ circulant matrices over $\mathbb{F}_p$ (see e.g. Sect 2.3 here).
In this paper we have written down explicit decompositions for the group of units in such rings as an abelian group. I imagine this would help to deal with your question too.
