Wang counterexamples lead to noncrossed products
HTML articles powered by AMS MathViewer
- by Eric S. Brussel PDF
- Proc. Amer. Math. Soc. 125 (1997), 2199-2206 Request permission
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
- S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408–420. MR 318216, DOI 10.1007/BF02764632
- 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
- Eric Brussel, Noncrossed products and nonabelian crossed products over $\mathbf Q(t)$ and $\mathbf Q((t))$, Amer. J. Math. 117 (1995), no. 2, 377–393. MR 1323680, DOI 10.2307/2374919
- Brussel, E.: Division algebras not embeddable in crossed products. Jour. Alg. 179 (1996), 631-655.
- Jürgen Neukirch, On solvable number fields, Invent. Math. 53 (1979), no. 2, 135–164. MR 560411, DOI 10.1007/BF01390030
- Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 674652
- I. Reiner, Maximal orders, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR 0393100
- 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
- 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
Additional Information
- Eric S. Brussel
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02143
- Email: brussel@math.harvard.edu
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2199-2206
- MSC (1991): Primary 16S35; Secondary 11R37
- DOI: https://doi.org/10.1090/S0002-9939-97-03725-8
- MathSciNet review: 1371116