Necessary and sufficient conditions on the order of a finite field ${\mathbb{F}}_q$ for the easy identification of primitive polynomials of degree 2

We present the necessary and sufficient conditions on the order $q$ of a finite field ${\mathbb{F}}_q$ such that every irreducible polynomial of the form $x^2+bx+c\in {\mathbb{F}}_q[x]$, with $b\mathrm{\ne }0$ and $c$ a primitive element of ${\mathbb{F}}_q$, is a primitive polynomial. As a by-product of this result, we also present a new infinite family of finite fields ${\mathbb{F}}_q$ for which it is easy, in a different way, to determine when an irreducible polynomial of degree two is primitive.