Article contents
On the symbol length of
$p$-algebras
Published online by Cambridge University Press: 22 May 2013
Abstract
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s) 2013
References
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