剩余类环上多项式的同余性质

Congruence properties of polynomials over residue class ring

摘要:

设$\mathbb{Z}/p^n\mathbb{Z}$ 是模$p^{n}$ 剩余类环，并且$U=\{f(x)\in{\mathbb Z}/p^{n} {\mathbb Z}[x]|f(a)\equiv 0~\pmod{p^{n}}, \forall a\in {\mathbb Z}\}$。 在本文中，我们证明了$U=\{f(x)\in{\mathbb Z}/p^{n}{\mathbb Z}[x]|f(a) \equiv 0~\pmod{p^{n}}, \forall a\in {\mathbb Z}\}$ 是自由生成的$\mathbb{Z}/p^n\mathbb{Z}$-模，进而我们给出了它的一组基， 并证明了商环$(\mathbb{Z}/p^n\mathbb{Z}[x])/U $是有限环， 且通过这组基确定了商环$(\mathbb{Z}/p^n\mathbb{Z}[x])/U $中的元素个数

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.