Congruence properties of polynomials over residue class ring

Abstract:

Suppose $\mathbb{Z}/p^n\mathbb{Z}$ is the residue ring of module $p^{n}$, and $U=\{f(x)\in{\mathbb Z}/p^{n}{\mathbb Z}[x]|f(a)\equiv 0~\pmod{p^{n}}, \forall a\in {\mathbb Z}\}$. In this thesis, we proved that $U=\{f(x)\in{\mathbb Z}/p^{n}{\mathbb Z}[x]|f(a)\equiv 0~\pmod{p^{n}}, \forall a\in {\mathbb Z}\}$ is a free generated $\mathbb Z/p^n \mathbb Z$-module, and then we get a set of bases of it, we also proved that the quotient ring $(\mathbb{Z}/p^n\mathbb{Z}[x])/U $ is a finite ring, then we can get the order of $(\mathbb{Z}/p^n\mathbb{Z}[x])/U $ through the bases of it.