Cyclicity conditions for division algebras of prime degree
HTML articles powered by AMS MathViewer
- by M. Mahdavi-Hezavehi and J.-P. Tignol PDF
- Proc. Amer. Math. Soc. 131 (2003), 3673-3676 Request permission
Abstract:
Let $D$ be a division algebra of prime degree $p$. A set of criteria is given for cyclicity of $D$ in terms of subgroups of the multiplicative group $D^*$ of $D$. It is essentially shown that $D$ is cyclic if and only if $D^*$ contains a nonabelian metabelian subgroup.References
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, Cambridge, 1983. MR 696937, DOI 10.1017/CBO9780511661907
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- M. Š. Huzurbazar, The multiplicative group of a division ring, Soviet Math. Dokl. 1 (1960), 433–435. MR 0120252
- M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of $\textrm {GL}_1(D)$, J. Algebra 241 (2001), no. 2, 720–730. MR 1843321, DOI 10.1006/jabr.2001.8782
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- J.-P. Tignol, Sur les décompositions des algèbres à division en produit tensoriel d’algèbres cycliques, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 126–145 (French). MR 657427
Additional Information
- M. Mahdavi-Hezavehi
- Affiliation: Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11365–9415, Tehran, Iran
- Email: mahdavih@sharif.edu
- J.-P. Tignol
- Affiliation: Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
- MR Author ID: 172580
- Email: tignol@math.ucl.ac.be
- Received by editor(s): June 19, 2002
- Received by editor(s) in revised form: July 15, 2002
- Published electronically: February 26, 2003
- Additional Notes: The first author thanks the Research Council of Sharif University of Technology for support. He also thanks Professor J.-P. Tignol for his hospitality during his stay at the Université Catholique de Louvain in May 2002.
The second author was partially supported by the National Fund for Scientific Research (Belgium). - Communicated by: Martin Lorenz
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3673-3676
- MSC (2000): Primary 16K20
- DOI: https://doi.org/10.1090/S0002-9939-03-06959-4
- MathSciNet review: 1998187