关于商环Z[i]/(a+bi)的若干性质

Some Properties of the Quotient Ring Z[i]/(a+bi)

本文给出了商环 Z(i)/(α+bi)与整数剩余类环 Z/(a^2+b^2)同构的充要条件,以及 Z(i)/(a+bi)作成域的充要条件,提出了 Z(i)/(p)与 Zp(x)/(x^2+1)及 Z(i)/(pi)的同构,並讨论了它作成域的充要条件.

This paper presents the-necessary and sufficient condition that the quotient ring Z[i]/(a+bi)be isomorphic with the congruence ring Z/(a^2+b^2)and also the necessary and sufficient condition that Z[i]/(a+bi)be a field.It proves that Z[i]/(p)is isomorphic both with Zp[x]/(x^2+1)and with Z[i]/(pi).Fina- lly,it discusses the necessary and sufficient condition of Z[i](p)being a field.