Some integer does not have a square root in $\mathbb{Z}_n$ if $n \geq 3$.

Question

Some integer does not have a square root in $\mathbb{Z}_n$ if $n \geq 3$.

I am having difficulty showing this. Can someone provide minimal guidance, or a small hint to point me in the right direction?

Answer 1

The map $S:x \mapsto x^2$ on $\mathbb{Z}_n$ is not injective because $S(x)=S(-x)$. Since $\mathbb{Z}_n$ is finite, $S$ cannot be surjective.