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Wang counterexamples lead
to noncrossed products

Author: Eric S. Brussel
Journal: Proc. Amer. Math. Soc. 125 (1997), 2199-2206
MSC (1991): Primary 16S35; Secondary 11R37
MathSciNet review: 1371116
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Abstract: Two famous counterexamples in algebra and number theory are Wang's counterexample to Grunwald's Theorem and Amitsur's noncrossed product division algebra. In this paper we use Wang's counterexample to construct a noncrossed product division algebra.

In the 30's, Grunwald's Theorem was used in the proof of a major result of class field theory, that all division algebras over number fields are (cyclic) crossed products. It is ironic that now Grunwald-Wang's Theorem is the decisive factor in a noncrossed product construction.

References [Enhancements On Off] (What's this?)

  • [A] S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408–420. MR 0318216
  • [AT] Emil Artin and John Tate, Class field theory, 2nd ed., Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. MR 1043169
  • [B] Eric Brussel, Noncrossed products and nonabelian crossed products over 𝐐(𝐭) and 𝐐((𝐭)), Amer. J. Math. 117 (1995), no. 2, 377–393. MR 1323680, 10.2307/2374919
  • [B2] Brussel, E.: Division algebras not embeddable in crossed products. Jour. Alg. 179 (1996), 631-655. CMP 96:06
  • [N] Jürgen Neukirch, On solvable number fields, Invent. Math. 53 (1979), no. 2, 135–164. MR 560411, 10.1007/BF01390030
  • [P] Richard S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR 674652
  • [R] I. Reiner, Maximal orders, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1975. London Mathematical Society Monographs, No. 5. MR 0393100
  • [S] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • [W] Shianghaw Wang, On Grunwald’s theorem, Ann. of Math. (2) 51 (1950), 471–484. MR 0033801

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Additional Information

Eric S. Brussel
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02143

Received by editor(s): April 12, 1995
Received by editor(s) in revised form: December 1, 1995
Additional Notes: The author’s research was supported in part by an Alfred P. Sloan Foundation Doctoral Dissertation Fellowship and by NSF Grant DMS-9100148
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society