Let $p$ be an odd prime and $h(x)=x^2+ax+1$ be an irreducible polynomial over $\mathbb{Z}_p$. I need to prove that the function

$$\Psi: \mathbb{Z}_p^2 \longrightarrow \mathbb{Z}_p, \quad (x,y)\mapsto x^2+y^2-x+y-axy$$

is surjective. I know that is true because the values in the image are in one-to-one correspondence with some conjugacy classes in some groups, but I would like to have an elementary proof of this fact, using the properties of polynomials over finite fields.

I tried to restrict to some suitable subset of the plane, like lines but I can not prove that the values the function takes when restricted to different lines cover all $\mathbb{Z}_p$.

Let $p$ be an odd prime and $h(x)=x^2+ax+1$ be an irreducible polynomial over the field $\mathbb{F}_p$. I need to prove that the function

$$\Psi: \mathbb{F}_p^2 \longrightarrow \mathbb{F}_p, \quad (x,y)\mapsto x^2+y^2-x+y-axy$$

is surjective. I know that is true because the values in the image are in one-to-one correspondence with some conjugacy classes in some groups, but I would like to have an elementary proof of this fact, using the properties of polynomials over finite fields.

I tried to restrict to some suitable subset of the plane, like lines but I can not prove that the values the function takes when restricted to different lines cover all $\mathbb{F}_p$.