Abstract
Let F be a field of characteristic p. We define and investigate nonassociative differential extensions of F and of a finite-dimensional central division algebra over F and give a criterium for these algebras to be division. As special cases, we obtain classical results for associative algebras by Amitsur and Jacobson. We construct families of nonassociative division algebras which can be viewed as generalizations of associative cyclic extensions of a purely inseparable field extension of exponent one or a central division algebra. Division algebras which are nonassociative cyclic extensions of a purely inseparable field extension of exponent one are particularly easy to obtain.
Article PDF
References
Amitsur, A.S.: Differential polynomials and division algebras. Ann. Math. 59(2), 245–278 (1954)
Amitsur, A.S.: Non-commutative cyclic fields. Duke Math. J. 21, 87–105 (1954)
Brown, C.: PhD Thesis University of Nottingham (in preparation)
Hoechsmann, K.: Simple algebras and derivations. Trans. Am. Math. Soc. 108, 1–12 (1963)
Jacobson, N.: Finite-Dimensional Division Algebras Over Fields. Springer, Berlin (1996)
Jacobson, N.: Abstract derivation and Lie algebras. Trans. Am. Math. Soc. 42(2), 206–224 (1937)
Kishimoto, K.: On cyclic extensions of simple rings. J. Faculty Sci. Hokkaido Univ. Ser. I(19), 74–85 (1966)
Lam, T.Y., Leung, K.H., Leroy, A., Matczuk, J.: Invariant and semi-invariant polynomials in skew polynomial rings. In: Ring Theory 1989 (Ramat Gan and Jerusalem, 1988/1989), pp. 247–261, Israel Mathematical Conference Proceedings, vol. 1. Weizmann, Jerusalem (1989)
Ore, O.: Formale theorie der linearen differentialgleichungen (Zweiter Teil) (German). J. Reine Angew. Math. 168, 233–252 (1932)
Petit, J.-C.: Sur certains quasi-corps généralisant un type d’anneau-quotient. Séminaire Dubriel. Algèbre et théorie des nombres 20, 1–18 (1966– 67)
Petit, J.-C.: Sur les quasi-corps distributifes à base momogène. C. R. Acad. Sci. Paris 266, 402–404 (1968)
Pumplün, S.: How to obtain lattices from \((f,\sigma ,\delta )\)-codes via a generalization of construction A. arXiv:1507.01491 [cs.IT]
Pumplün, S., Steele, A.: Fast-decodable MIDO codes from nonassociative algebras. Int. J. Inf. Coding Theory (IJICOT) 3(1), 15–38 (2015)
Schafer, R.D.: An Introduction to Nonassociative Algebras. Dover Publ. Inc., New York (1995)
Steele, A.: Nonassociative cyclic algebras. Isr. J. Math. 200(1), 361–387 (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Pumplün, S. Nonassociative Differential Extensions of Characteristic \(\varvec{p}\) . Results Math 72, 245–262 (2017). https://doi.org/10.1007/s00025-017-0656-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0656-x