Generalized Artin-Schreier polynomials
Natalio H. Guersenzvaig, Fernando Szechtman

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 be a field of prime characteristic containing as a subfield. We refer to as a generalized Artin-Schreier polynomial. Suppose that is irreducible and let be the companion matrix of . Then has such highly unusual properties that any such that 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 , 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Coding theory and cryptography · Algebraic structures and combinatorial models
