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On the symbol length of $p$-algebras

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  22 May 2013

Mathieu Florence
Mathieu Florence*
Affiliation:
Equipe de Topologie et Géométrie Algébriques, Institut de Mathématiques de Jussieu, 4 place Jussieu, 75005 Paris, France email mathieu.florence@gmail.com
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Abstract

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The main result of this paper states that if $k$ is a field of characteristic $p\gt 0$ and $A/ k$ is a central simple algebra of index $d= {p}^{n} $ and exponent ${p}^{e} $, then $A$ is split by a purely inseparable extension of $k$ of the form $k( \sqrt[{p}^{e} ]{{a}_{i} }, i= 1, \ldots , d- 1)$. Combining this result with a theorem of Albert (for which we include a new proof), we get that any such algebra is Brauer equivalent to the tensor product of at most $d- 1$ cyclic algebras of degree ${p}^{e} $. This gives a drastic improvement upon previously known upper bounds.

Keywords

central simple algebrasSeveri–Brauer varieties

MSC classification

Secondary: 20G15: Linear algebraic groups over arbitrary fields
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013 , pp. 1353 - 1363
Copyright
© The Author(s) 2013 

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