Exercise 1.4.4 (If $n$ is not a prime then $\mathbb{Z}/n \mathbb{Z}$ is not a field.)

Question

Show that if $n$ is not a prime then $\mathbb{Z}/n \mathbb{Z}$ is not a field.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} 
Let $n \in \Z$.

If $n = 0$, then $\Z/0 \, \Z \simeq \Z$ is not a field.

If $n = 1$, then $\Z/1\,  \Z \simeq \{0\}$ is not a field (there is no element $1 
e 0$).

Since $\Z/(-n)\Z = \Z/n\Z$, we may suppose that $n\geq 2$. If $n$ is not prime, then it is composite, so there are integers $a, b$ such that $1 < a < n$ and $1 < b < n$ and $n = ab$. Then 
$$\overline{0} = \overline{n} = \overline{a} \overline{b}, 	ext{where } \overline{a} 
e 0, \overline{b} 
e \overline{0}.
$$
Since $\Z/n\Z$ has divisors of $0$, it cannot be a field: otherwise $\overline{a}  
e \overline{0}$ has an inverse $\overline{a} ^{-1}$, so 
$$\overline{0} = \overline{a}^{-1}\cdot \overline{0} = \overline{a}^{-1}(\overline{a} \overline{b}) = \overline{b},$$
thus $\overline{b} = \overline{0} $: this is a contradiction, so $\Z/n \Z$ is not a field if $n$ is not a prime.
\end{proof}

Exercise 1.4.4 (If $n$ is not a prime then $\mathbb{Z}/n \mathbb{Z}$ is not a field.)

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Exercise 1.4.4 (If $n$ is not a prime then $\mathbb{Z}/n \mathbb{Z}$ is not a field.)

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