Assuming $q$ is a power of an odd prime $p$ I would like to find a necessary and sufficient condition on $a,b$ such that $P(x)=x^2 +ax+b$ is irreducible over $\mathbb F_q$.
I thought about enforcing the discriminant of $P(x)$ to be quadratic non residue, however is it really necessary and sufficient in finite fields?.
TNX in advance.