Semi-characteristic polynomials, {\phi}-modules and skew polynomials

TL;DR

This paper introduces semi-characteristic polynomials for semi-linear maps, providing new tools to analyze skew and linearized polynomials over finite fields, including algorithms for splitting fields and factorizations.

Contribution

It develops the concept of semi-characteristic polynomials and applies it to study skew polynomials, offering algorithms for splitting fields, Galois actions, and factorizations.

Findings

Algorithm to compute the splitting field of linearized polynomials

Method to determine the Galois action on the splitting field

Procedure to count and enumerate all factorizations of 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.