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

Question

I read book of Dummit and Foot Abstract algebra. I need some help with the following question.

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

I know definitions of field and group $\mathbb{Z}/n\mathbb{Z}.$ I have no idea how to start it.

Answer 1

Hint:

If $n$ is not prime, use the fact that $n=n_1n_2$, where $1<n_1,n_2<n$ to find divisors of $0$ in $\mathbb Z/n\mathbb Z$.

Answer 2

Field definition states closure under addition and multiplication.

If $\mathbb Z\setminus{n \mathbb Z}$ is a field than:

For $n_1,n_2 \in \mathbb Z \setminus{n \mathbb Z}$ than field definition requires that $n_1 * n_2 \in \mathbb Z \setminus{n \mathbb Z}$

Write $n$ as $n_1 * n_2$ while $n_1,n_2 > 1$ (therefore $n$ is not prime), assume negative that $\mathbb Z\setminus{n \mathbb Z}$ is a field and show the contradiction.