# When is a quotient ring of the Gaussian integers an integral domain?

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?

-

1
Why do you post two questions within 5 minutes with two different user names? – Phira Dec 5 '12 at 2:43

1
when n is a gaussian prime, so 19 and 17 (because they equal 3 mod 4). – user51427 Dec 5 '12 at 2:44

## 3 Answers

$Z[i]/n Z[i]$ is an integral domain iff $n$ is a Gaussian prime.

No composite integer $n$ can be a Gaussian prime.

$2$ is not a Gaussian prime because $2=(1+i)(1-i)=(1+i)^2(-i)$.

Primes $p$ that are the sum of two squares are not Gaussian primes, because $p=a^2+b^2=(a+bi)(a-bi)$. So $p=2$ and $p\equiv 1\bmod 4$ are not Gaussian primes.

This leaves us with the primes that are not the sum of two squares, which are exactly those that are congruent to $3 \bmod 4$. These are indeed Gaussian primes.

-

A small start: $(3+2i)(3-2i)$ is divisible by $13$.

-

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.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 19, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8982937335968018, "perplexity_flag": "head"}