Semi-characteristic polynomials, {\phi}-modules and skew polynomials
Abstract
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.
Cite
@article{arxiv.1105.4083,
title = {Semi-characteristic polynomials, {\phi}-modules and skew polynomials},
author = {Jérémy Le Borgne},
journal= {arXiv preprint arXiv:1105.4083},
year = {2011}
}