Let p be an odd prime, q the smallest quadratic non residue (mod p). Prove q is prime.

Question

So I have this problem;

Let p be an odd prime and let q be the smallest positive integer which is a quadratic non residue (mod p). Prove q is a prime.

So what I know is that, since q is the smallest positive integer which is a quadratic non residue (mod p) then Legendre symbol (q|p) = -1 = q^((p-1)/2) (mod p). But I'm not sure if this is the correct direction with what I am doing since I can't concluding anything (obviously) from this.

Answer 1

Hint: if $q$ is composite, it can be written as $ab$ where $a < q$ and $b < q$. Can $a$ and $b$ both be quadratic residues mod $p$?

Answer 2

For the sake of contradiction we assume that q is composite. That is, q can be written in the form q = ab, where a,b∈Z with a,b < q. By rules of Legendre Symbols, we know that (q|p) = (ab|p) = (a|p) ∙ (b|p) = -1. That is one of (a|p), (b|p) = -1, and the other is equivalent to 1. That is whichever is equivalent to -1 is a quadratic non residue (mod p) smaller than q, a contradiction. Therefore, q must be prime.