## Symbol length in the Brauer group of a field

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- by Eliyahu Matzri PDF
- Trans. Amer. Math. Soc.
**368**(2016), 413-427 Request permission

## 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.## References

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## Additional Information

**Eliyahu Matzri**- Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
- 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. - © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3413868