Is it possible to construct an irreducible polynomial $f$ over $\mathbb{F}_{q}$ such that $f(x)$ is a non-square for any $x \in \mathbb{F}_{q}$?
I can prove the existence of irreducible polynomials (Euclid's argument), and I can construct polynomials with no square values (for example by Lagrange interpolation through non-squares), but satisfying these 2 conditions feels difficult.
Motivation: I am trying to construct an hyper-elliptic curve $y^2 = f(x)$ with no rational points.
EDIT: Can you construct such an $f$ that will not be constant on the ground field $\mathbb{F}_{q}$?