The eigenspaces of twisted polynomials over cyclic field extensions

TL;DR

This paper investigates the eigenspaces of skew polynomials over cyclic field extensions using nonassociative algebra, providing bounds, explicit computations, and reducibility conditions for these polynomials.

Contribution

It introduces a novel approach employing nonassociative algebra to analyze eigenspaces of skew polynomials over cyclic extensions, including bounds and reducibility criteria.

Findings

Lower bounds on eigenspace dimensions are established.

Explicit eigenspace computations are provided for certain cases.

Conditions for reducibility of polynomials are identified.

Abstract

Let $K$ be a field and $\sigma$ an automorphism of $K$ of order $n$.Employing a nonassociative algebra, we study the eigenspace of a bounded skew polynomial $f\in K[t;\sigma]$. We mainly treat the case that $K/F$ is a cyclic field extension of degree $n$ with Galois group generated by $\sigma$. We obtain lower bounds on the dimension of the eigenspace, and compute it in special cases as a quotient algebra. Conditions under which a monic polynomial $f\in F[t]\subset K[t;\sigma]$ is reducible are obtained in special cases.