<abstract xmlns="http://www.w3.org/1999/xhtml">

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

The eigenspaces of twisted polynomials over cyclic field extensions

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"The eigenspaces of twisted polynomials over cyclic field extensions"}

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]....

The eigenspaces of twisted polynomials over cyclic field extensions

Published Online: Feb 04, 2023

Page range: 221 - 240

Received: Jun 17, 2022

Accepted: Sep 20, 2022

DOI: https://doi.org/10.2478/auom-2023-0012

Keywords
Skew polynomial ring, reducible skew polynomials, eigenspace, nonassociative algebra

© 2023 Adam Owen et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The eigenspaces of twisted polynomials over cyclic field extensions

Published Online: Feb 04, 2023

Page range: 221 - 240

Received: Jun 17, 2022

Accepted: Sep 20, 2022

DOI: https://doi.org/10.2478/auom-2023-0012

KeywordsSkew polynomial ring, reducible skew polynomials, eigenspace, nonassociative algebra

© 2023 Adam Owen et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Skew polynomial ring, reducible skew polynomials, eigenspace, nonassociative algebra