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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?)

  • [A] A. Adrian Albert, Structure of algebras, Revised printing. American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. MR 0123587
  • [B] Richard Brauer, über den Index und den Exponenten von Divisionsalgebren, Tôhoku Math. J. 37 (1933), 77-87.
  • [R] Lawrence Risman, Subalgebras of division algebras, Ph.D. Dissertation, Harvard University, Cambridge Mass., 1973.
  • [S] Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968 (French). Deuxième édition; Publications de l’Université de Nancago, No. VIII. MR 0354618
  • [Sc] Murray M. Schacher, Subfields of division rings. I, J. Algebra 9 (1968), 451–477. MR 0227224

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Article copyright: © Copyright 1975 American Mathematical Society