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Wang counterexamples lead to noncrossed products

Author(s): Eric S. Brussel
Journal: Proc. Amer. Math. Soc. 125 (1997), 2199-2206.
MSC (1991): Primary 16S35; Secondary 11R37
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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:

[A]: Amitsur, S.A.: On central division algebras. Israel J. Math. 12 (1972), 408-422. MR 47:6763
[AT]: Artin, E., Tate, J.: Class Field Theory, Addison-Wesley, Reading, Mass., 1967. MR 91b:11129
[B]: Brussel, E.: Noncrossed products and nonabelian crossed products over Q(t) and Q((t)). Amer. Jour. Math. 117 (1995), 377-393. MR 96a:16014
[B2]: Brussel, E.: Division algebras not embeddable in crossed products. Jour. Alg. 179 (1996), 631-655. CMP 96:06
[N]: Neukirch, J: On solvable number fields. Invent. Math. 53 (1979), 135-164. MR 81e:12009
[P]: Pierce, R. S.: Associative Algebras, Springer-Verlag, New York, 1982. MR 84c:16001
[R]: Reiner, I.: Maximal Orders, Academic Press, London, 1975. MR 52:13910
[S]: Serre, J.-P.: Local Fields, Springer-Verlag, New York, 1979. MR 82e:12016
[W]: Wang, S.: On Grunwald's theorem. Ann. of Math. (2) 51 (1950), 471-484. MR 11:489h

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Additional Information:

Eric S. Brussel
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02143
Email: brussel@math.harvard.edu

DOI: 10.1090/S0002-9939-97-03725-8
PII: S 0002-9939(97)03725-8
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 of article: Copyright 1997, American Mathematical Society


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