Regarding the ring $\mathbb{Z}[X]/\Phi_n(X)$ for the $n$'th cyclotomic polynomial $\Phi_n$, I find much literature on the nature of this ring taken modulo a prime $q$, that is literature on the splitting of prime ideals in Galois extensions. But I don't understand why the analysis only concerns the splitting of prime ideals, not mentioning the splitting of non-prime ideals.
I find nothing regarding the case when $q$ is not prime. If $q$ has prime decomposition with multiplicities all $1$ then the Chinese Remainder Theorem reduces analysis back to the prime case. But what about the case when $q$ is or contains a prime power? What is the nature of the ring $\mathbb{Z}_{p^e}[X]/\Phi_n(X)$ for a prime $p$ and power $e>1$? In the case that $\Phi_n(X)$ is irreducible modulo $p$ we have a Galois ring. Where can I find literature on the more general case when $\Phi_n$ may split modulo $p$?
I'm particularly interested in the cases $p=2$, and/or $n=2^k$. As a specific case, how do I calculate how $\Phi$ splits into irreducible in the ring $$ \mathbb{Z}_{2^e}[X]/\Phi_{2^k}(X) $$ with, say, $e > k$, e.g. $e=32$, $k=10$.