<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

Online veröffentlicht: 04. Feb. 2023

Seitenbereich: 221 - 240

Eingereicht: 17. Juni 2022

Akzeptiert: 20. Sept. 2022

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

Schlüsselwörter
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

Online veröffentlicht: 04. Feb. 2023

Seitenbereich: 221 - 240

Eingereicht: 17. Juni 2022

Akzeptiert: 20. Sept. 2022

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

SchlüsselwörterSkew 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.

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