Generalized Artin-Schreier polynomials

Let $F$ be a field of prime characteristic $p$ containing $F_{p^n}$ as a subfield. We refer to $q(X)=X^{p^n}-X-a\in F[X]$ as a generalized Artin-Schreier polynomial. Suppose that $q(X)$ is irreducible and let $C_{q(X)}$ be the companion matrix of $q(X)$. Then $ad\, C_{q(X)}$ has such highly unusual properties that any $A\in{\mathfrak{ gl}}(m)$ such that $ad\, A$ has like properties is shown to be similar to the companion matrix of an irreducible generalized Artin-Schreier polynomial. We discuss close connections with the decomposition problem of the tensor product of indecomposable modules for a 1-dimensional Lie algebra over a field of characteristic $p$, the problem of finding an explicit primitive element for every intermediate field of the Galois extension associated to an irreducible generalized Artin-Schreier polynomial, and the problem of finding necessary and sufficient conditions for the irreducibility of a family of polynomials.