I am struggling to understand a couple of statements in a cryptography-related paper. I think I lack some maths background. Can you help me understand it ?
Here are the statements:
We consider the ring $\mathbb{Z}[X]/(\Phi_m(X))$, where $\Phi_m(X)$ is the $m$-th cyclotomic polynomial. Suppose the factorization of $\Phi_m(x) \bmod p^r$ for a prime power $p^r$ is $\Phi_m(X)=F_1(X)...F_k(X)$, where each $F_i$ has the same degree $d$.
We define $E=\mathbb{Z}[X]/(p^r, F_1(X))$ and let $\zeta$ be the residue class of $X$ in $E$, which is a principal $m$-th root of unity, so that $E=\mathbb{Z}/(p^r)[\zeta]$.....
(extracted from Bootstrapping for HElib, Halevi and Shoup, 2015).
The sentence that I don't understand is the last one. How come $\zeta$ can be the residue class of $X$ in $E$ and why is it a $m$-th root of unity ??
Besides, I don't understand the notation $E=\mathbb{Z}/(p^r)[\zeta]$.
Any explanation would be helful. Many thanks.