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Transactions of the American Mathematical Society

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Symbol length in the Brauer group of a field


Author: Eliyahu Matzri
Journal: Trans. Amer. Math. Soc. 368 (2016), 413-427
MSC (2010): Primary 12G05, 16K50, 17A35; Secondary 19D45, 19C30
DOI: https://doi.org/10.1090/tran/6326
Published electronically: April 15, 2015
MathSciNet review: 3413868
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Abstract: We bound the symbol length of elements in the Brauer group of a field $ K$ containing a $ C_m$ field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a $ C_m$ field $ F$. In particular, for a $ C_m$ field $ F$, we show that every $ F$ central simple algebra of exponent $ p^t$ is similar to the tensor product of at most $ \operatorname {len}(p^t,F)\leq t(p^{m-1}-1)$ symbol algebras of degree $ p^t$. We then use this bound on the symbol length to show that the index of such algebras is bounded by $ (p^t)^{(p^{m-1}-1)}$, which in turn gives a bound for any algebra of exponent $ n$ via the primary decomposition. Finally for a field $ K$ containing a $ C_m$ field $ F$, we show that every $ F$ central simple algebra of exponent $ p^t$ and degree $ p^s$ is similar to the tensor product of at most $ \operatorname {len}(p^t,p^s,K)\leq \operatorname {len}(p^t,L)$ symbol algebras of degree $ p^t$, where $ L$ is a $ C_{m+\operatorname {ed}_L(A)+p^{s-t}-1}$ field.


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Additional Information

Eliyahu Matzri
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel

DOI: https://doi.org/10.1090/tran/6326
Received by editor(s): October 16, 2013
Received by editor(s) in revised form: November 7, 2013
Published electronically: April 15, 2015
Additional Notes: The author thanks Daniel Krashen, Andrei Rapinchuk, Louis Rowen, David Saltman and Uzi Vishne for all their help, time and support.
This work was partially supported by the BSF, grant number 2010/149.
Article copyright: © Copyright 2015 American Mathematical Society

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