Generalized Artin-Schreier polynomials

TL;DR

This paper studies the properties of generalized Artin-Schreier polynomials over fields of prime characteristic, revealing their unique algebraic features and connections to module decomposition, Galois theory, and polynomial irreducibility.

Contribution

It characterizes matrices with properties similar to companion matrices of irreducible generalized Artin-Schreier polynomials and explores their links to module decomposition and Galois extensions.

Findings

Companion matrices of irreducible generalized Artin-Schreier polynomials have unique properties.

Matrices with similar properties are similar to such companion matrices.

Connections established with module decomposition and Galois theory problems.

Abstract

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.