  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.
