TITLE: When is a quotient ring of the Gaussian integers an integral domain?
QUESTION [1 upvotes]: Let $\mathbb Z[i]$ denote the ring of Gaussian integers. For which of the following values of $n$ is the quotient ring $\mathbb Z[i]/n \mathbb Z[i]$ an integral domain?
$2,13,19,7$
How can I solve the problem?

REPLY [0 votes]: Hint $\rm\ \, R = \Bbb Z[{\it i}\,]\cong \Bbb Z[x]/(x^2\!+\!1)\:$ so $\rm\:R/2 \cong \Bbb Z_2[x]/(x^2\!+\!1) \cong \Bbb Z_2[x]/(x\!+\!1)^2\:$ is not a domain. Similarly, we easily compute that $\rm R/13 \cong  \Bbb Z_{13}[x]/(x^2\!+\!1) \cong \Bbb Z_{13}[x]/(x\!-\!5)(x\!+\!5)\:$ is not a domain.
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