剩余类环上多项式的同余性质

设Z/p~nZ是模p~n剩余类环.本文证明了U={f(x)∈Z/p~nZ[x]|f(a)≡0(modp~n),■a∈Z}是自由生成的Z/p~nZ-模,给出了它的一组基,还证明了商环(Z/p~nZ[x])/U是有限环,并通过这组基确定了商环(Z/p~nZ[x])/U中的元素个数.

Congruence properties of polynomials over residue class ring

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