Stability: index and order in the Brauer group
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- by Lawrence J. Risman PDF
- Proc. Amer. Math. Soc. 50 (1975), 33-39 Request permission
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
- A. Adrian Albert, Structure of algebras, American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. Revised printing. MR 0123587 Richard Brauer, über den Index und den Exponenten von Divisionsalgebren, Tôhoku Math. J. 37 (1933), 77-87. Lawrence Risman, Subalgebras of division algebras, Ph.D. Dissertation, Harvard University, Cambridge Mass., 1973.
- Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618
- Murray M. Schacher, Subfields of division rings. I, J. Algebra 9 (1968), 451–477. MR 227224, DOI 10.1016/0021-8693(68)90015-X
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 50 (1975), 33-39
- MSC: Primary 12A90; Secondary 12G05, 16A16
- DOI: https://doi.org/10.1090/S0002-9939-1975-0379442-4
- MathSciNet review: 0379442