How do I guarantee the existence of a quadratic non-residue in $\mathbb{F}_p$

Question

I need to find $q \in \mathbb{F}_p$ such that $x^2+q$ has no roots in $\mathbb{F}_p$. How can I guarantee such $q$ exists? Only ways I have heard of involve number-theoretic methods we don't know. I was wondering if there was any straightforward way to see this.

Answer 1

If $p \ge 3$, the map $x \mapsto x^2$ is not injective because $(-1)^2=1^2=1$.

Since $\Bbb F_p$ is a finite set, that map can't be surjective.