  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.
