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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Stability: index and order in the Brauer group


Author: Lawrence J. Risman
Journal: Proc. Amer. Math. Soc. 50 (1975), 33-39
MSC: Primary 12A90; Secondary 12G05, 16A16
MathSciNet review: 0379442
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Abstract: A field is stable if for every division algebra $ A$ in its Brauer group order of $ A$ = index of $ A$. Index and order in the Brauer group of a field $ F$ with discrete valuation and perfect residue class field $ K$ are calculated. Division algebras with specified order and index are constructed.

For $ F$ complete, necessary and sufficient conditions for the stability of $ F$ are given in terms of the Brauer group of $ K$. These results follow. A finite extension of a stable field need not be stable. The power series field $ K((x))$ is stable for $ K$ a local field. $ K((x))$ and $ K(x)$ are not stable for $ K$ a global field.


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

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  • [R] Lawrence Risman, Subalgebras of division algebras, Ph.D. Dissertation, Harvard University, Cambridge Mass., 1973.
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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0379442-4
PII: S 0002-9939(1975)0379442-4
Article copyright: © Copyright 1975 American Mathematical Society