Generalized Artin–Schreier polynomials

Let F be a field of prime characteristic p containing 𝔽pn as a subfield. We refer to q(X) = Xpn - X - a ∈ F[X] as a generalized Artin–Schreier polynomial. Suppose that q(X) is irreducible and let Cq(X) be the companion matrix of q(X). Then ad Cq(X) has such highly unusual properties that any A ∈ 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 one-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.