Irreducible skew polynomials over domains

C. Brown; S. Pumplün

doi:10.2478/auom-2021-0035

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Irreducible skew polynomials over domains

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 29 (2021): Issue 3 (November 2021)

By: C. Brown and S. Pumplün

Open Access

|Nov 2021

Abstract

Let S be a domain and R = S[t; σ, δ] a skew polynomial ring, where σ is an injective endomorphism of S and δ a left σ -derivation. We give criteria for skew polynomials f ∈ R of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider f(t) = t^m − a ∈ R.

References

[1] G. Benkart, S. A. Lopes, M. Ondrus, A parametric family of subalgebras of the Weyl algebra I. Structure and automorphisms. Trans. Amer. Math. Soc. 367 (3) (2015), 1993-2021.10.1090/S0002-9947-2014-06144-8
Open DOI Search in Google Scholar Back to article
[2] G. Benkart, S. A. Lopes, M. Ondrus, A parametric family of subalgebras of the Weyl algebra II. Irreducible modules. Online at arXiv:1212.1404 [math.RT]
Search in Google Scholar Back to article
[3] G. Benkart, S. A. Lopes, M. Ondrus, Derivations of a parametric family of subalgebras of the Weyl algebra. J. Algebra 424 (2015), 46-97.
Search in Google Scholar Back to article
[4] N. Boubaki, Éléments de mathématique. Fasc. XXIII. Algèbre; Chapitre 8, modules e anneaux semi-simples. Hermann, 1973.
Search in Google Scholar Back to article
[5] C. Brown, S. Pumplün, How a nonassociative algebra reflects the properties of a skew polynomial. Glasgow Mathematical Journal (Nov. 2019) https://doi.org/10.1017/S001708951900047810.1017/S0017089519000478
Open DOI Search in Google Scholar Back to article
[6] P. M. Cohn, “Skew fields. Theory of general division rings.” Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1995.10.1017/CBO9781139087193
Open DOI Search in Google Scholar Back to article
[7] C. Cook, K. Price, Factorization in quantized Weyl algebras. Preprint, University of Wisconsin Oshkosh August 24, 2005
Search in Google Scholar Back to article
[8] R. Coulibaly, K. Price, Factorization in quantum planes. Missouri J. Math. Sci. 18 (3)(2006), 197-205.10.35834/2006/1803197
Open DOI Search in Google Scholar Back to article
[9] J. Gòmez-Torrecillas, F. J. Lobillo, G. Navarro, Computing the bound of an Ore polynomial. Applications to factorization. J. Symbolic Comp. 2018, online at https://doi.org/10.1016/j.jsc.2018.04.01810.1016/j.jsc.2018.04.018
Open DOI Search in Google Scholar Back to article
[10] K. Goodearl, Prime ideals in skew polynomial rings and quantized Weyl algebras. J. Algebra 150 (2) (1992), 324-377.10.1016/S0021-8693(05)80036-5
Open DOI Search in Google Scholar Back to article
[11] C. Holtz, K. Price, Normal quadratics in Ore extensions, quantum planes, and quantized Weyl algebras. Acta Applicandae Mathematicae 108 (1) (2009), 73-81.10.1007/s10440-008-9371-7
Open DOI Search in Google Scholar Back to article
[12] N. Jacobson, “Finite-dimensional division algebras over fields.” Springer Verlag, Berlin-Heidelberg-New York, 1996.10.1007/978-3-642-02429-0
Open DOI Search in Google Scholar Back to article
[13] T. Y. Lam, A. Leroy, Vandermonde and Wroskian matrices over division rings. J. Alg. 19(2)(1988), 308-336.10.1016/0021-8693(88)90063-4
Open DOI Search in Google Scholar Back to article
[14] J.-C. Petit, Sur certains quasi-corps généralisant un type d’anneau-quotient. Séminaire Dubriel. Algèbre et théorie des nombres 20 (1966-67), 1-18.
Search in Google Scholar Back to article
[15] O. Ore, Theory of noncommutative polynomials. Annals of Math. 34 (3) (1933), 480-508.10.2307/1968173
Open DOI Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/auom-2021-0035 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 75 - 89

Submitted on: Apr 22, 2021

Accepted on: Apr 30, 2021

Published online: Nov 23, 2021

Published in print: Nov 1, 2021

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Keywords:

Skew polynomials,

irreducible polynomials

MSC:

Primary: 16S36

Related subjects:

Mathematics,

General mathematics

© 2021 C. Brown, S. Pumplün, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 29 (2021): Issue 3 (November 2021)