Generalized Artin-Schreier polynomials

2013/06/17 by Natalio H. Guersenzvaig, Guersenzvaig, Natalio H., Fernando Szechtman +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Coding theory and cryptography #math.RA #msc:12F10 #msc:12F20 #msc:15A21 #msc:17B50

paper · pdf · doi:10.48550/arxiv.1306.3967

arxiv created 2014/08/19 · arxiv updated 2014/08/20

Abstract

Let F be a field of prime characteristic p containing F_pⁿ as a subfield. We refer to q(X)=X^pn-X-a∈ 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∈\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.

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• On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras

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