The eigenspaces of twisted polynomials over cyclic field extensions

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.