Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Semidirect product division algebras

  • 69 Accesses

  • 4 Citations


SupposeD is a division algebra of degreep over its centerF, which contains a primitivep-root of 1. Also supposeD has a maximal separable subfield overF whose Galois group is the semidirect product of the cyclic groupsC p C q , whereq=2, 3, 4, or 6 and is relatively prime top (In particular this is the case whenp is prime ≤7 andD has a maximal separable subfield whose Galois group is solvable.) ThenD is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.

This is a preview of subscription content, log in to check access.


  1. [B] K. Brown,Cohomology of Groups, Springer-Verlag, New York-Heidelberg-Berlin, 1982.

  2. [CTS] J.-L. Colliot-Thélène and J.-J. Sansuc,La R-équivalence sur les tores, Annales Scientifiques de l'École Normale Supérieure4 (1977), 175–230.

  3. [DI] F. DeMeyer and E. Ingraham,Separable algebras over commutative rings, Lecture Notes in Mathematics181. Springer-Verlag, Berlin-Heidelberg-New York, 1971.

  4. [D] P. K. Draxl,Skew Fields, Cambridge University Press, Cambridge, 1983.

  5. [Fi] E. Fischer,Die Isomorphie der Invariantenkorper der endlichen Abel'schen Gruppen linearen transformationen, Gott. Nachr. (1915), 77–80.

  6. [Fo] T. Ford,Division algebras that ramify only along a singular plane cubic curve, New York Journal of Mathematics1 (1995), 178–183, http://nyjm.albany.edu:8000/j/v1/ford.html.

  7. [BAI] N. Jacobson,Basic Algebra I, Freeman, San Francisco, 1974.

  8. [KO] M. A. Knus and M. Ojanguren,Theorie de la Descente et Algèbres d'Azumaya, Lecture Notes in Mathematics389, Springer-Verlag, Berlin, 1974.

  9. [MT] P. Mammone and J.-P. Tignol,Dihedral algebras are cyclic, Proceedings of the American Mathematical Society101 (1987), 217–218.

  10. [Mi] J. S. Milne,Etale Cohomology, Princeton University Press, Princeton, 1980.

  11. [OS] M. Orzech and C. Small,The Brauer groups of commutative rings, Marcel Dekker, New York, 1975.

  12. [RT] S. Rosset and J. Tate,A reciprocity law for generalized traces, Commentarii Mathematici Helvetici58 (1983), 38–47.

  13. [RS] L. H. Rowen and D. Saltman,Dihedral algebras are cyclic, Proceedings of the American Mathematical Society84 (1981), 162–164.

  14. [S] D. J. Saltman,Generic Galois extensions and problems in field theory, Advances in Mathematics43 (1982), 250–283.

  15. [S2] D. J. Saltman,Azumaya algebras with involution, Journal of Algebra52 (1978), 526–539.

  16. [Se] J.-P. Serre,Local Fields Springer-Verlag, New York, 1979.

  17. [Ti1] J.-P. Tignol,Galois' Theory of Algebraic Equations, Longman Scientific and Technical, Essex, England, 1988.

  18. [Ti2] J.-P. Tignol,Metacyclic division algebras of degree 5, inRing Theory 1989 (L. H. Rowen, ed.), Israel Mathematical Conference Proceedings, Weizmann Science Press of Israel, Jerusalem, 1989.

Download references

Author information

Correspondence to Louis H. Rowen.

Additional information

Research supported in part by US-Israel Binational Science Foundation grant #92-00255. The second author is grateful for support under NSF grant DMS-9400650. Also, the authors thank the referee for several very helpful suggestions.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rowen, L.H., Saltman, D.J. Semidirect product division algebras. Israel J. Math. 96, 527–552 (1996). https://doi.org/10.1007/BF02937322

Download citation


  • Direct Summand
  • Galois Group
  • Prime Power
  • Division Algebra
  • Semidirect Product