The eigenspaces of twisted polynomials over cyclic field extensions

Abstract

Let K be a field and σ an automorphism of K of order n. Employing a nonassociative algebra, we study the eigenspace of a bounded skew polynomial f ∈ K[t; σ]. We mainly treat the case that K/F is a cyclic field extension of degree n with Galois group generated by σ.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 ∈ F [t] ⊂ K[t; σ] is reducible are obtained in special cases.

Keywords

• skew polynomial ring
• reducible skew polynomials
• eigenspace
• nonassociative algebra
• primary: 17a35
• secondary: 17a60, 17a36, 16s36