Let $k$ be an algebraically closed field of positive characteristic $p$. Suppose that $F$ is a monic irreducible polynomial over $k[x]$ of degree $p$ whose discriminant is an element of $k$. Is then $F$ necessarily of the form $$F=T^p+aT+f(x),$$ for some $a\in k$ and $f(x)\in k[x]$?
I'm interested in this because I'm looking at degree-$p$ étale covers of the affine line $\mathbb{A}^1_k$. Of course Artin-Schreier covers like the above provide examples, and I know that there are non-Galois degree-$p$ étale covers of $\mathbb{A}_k^1$ as well. But are there any non-Artin-Schreier covers given by a polynomial $F$ as above with constant discriminant? If I am not mistaken, this means that the covering curve is a smooth plane curve.