show that $Z_{p}$ are prime fields,where $p$ is prime numbers.
maybe this problem is old,But I look for some book,and can't find it,someone know which book have this problem proof?
because I know this if $z_{n}$ is a field,if and only if $n$ is prime, see this book:page 2
Thank you
If $\overline{r}\in \mathbb{Z}_p$ and $\overline{r} \neq 0 $ then $\gcd(r,p) = 1 $ and thus there exist $m , n \in \mathbb{Z} $ such that $$mr + np = 1$$
Thus $\overline{m} \cdot \overline{r} = \overline{1} $ in $\mathbb{Z}_p$ , $\overline{r}$ is invertible and so $\mathbb{Z}_p$ is a field.
If $n$ is not prime, then $n = ab $ with $a, b \neq n $ and so $$\overline{a} \cdot \overline{b} = 0 \ \ \ \overline{a} \neq 0, \ \ \overline{b} \neq 0$$ Thus $\mathbb{Z}_n$ is not even a domain.