2011/05/20 by Jérémy Le Borgne, Borgne, Jérémy Le
Computer Science · Mathematics · #Advanced Topics in Algebra #Coding theory and cryptography #FOS: Mathematics #Polynomial and algebraic computation #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1105.4083
arxiv created 2011/05/20 · openalex publication_date 2011/05/20 · arxiv updated 2011/05/23 · openalex created_date 2022/09/11 · openalex updated_date 2026/07/28
We introduce the notion of semi-characteristic polynomial for a semi-linear map of a finite- dimensional vector space over a field of characteristic p. This polynomial has some properties in common with the classical characteristic polynomial of a linear map. We use this notion to study skew polynomials and linearized polynomials over a finite field, giving an algorithm to compute the splitting field of a linearized polynomial over a finite field and the Galois action on this field. We also give a way to compute the optimal bound of a skew polynomial. We then look at properties of the factorizations of skew polynomials, giving a map that computes several invariants of these factorizations. We also explain how to count the number of factorizations and how to find them all.