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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Brauer factor sets and simple algebras


Author: Louis H. Rowen
Journal: Trans. Amer. Math. Soc. 282 (1984), 765-772
MSC: Primary 16A39; Secondary 12E15, 16A38
MathSciNet review: 732118
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Abstract: It is shown that the Brauer factor set $ ({c_{ijk}})$ of a finite-dimensional division algebra of odd degree $ n$ can be chosen such that $ {c_{iji}} = {c_{iij}} = {c_{jii}} = 1$ for all $ i,j$ and $ {c_{ijk}} = c_{kji}^{ - 1}$. This implies at once the existence of an element $ a \ne 0$ with $ {\text{tr}}(a) = {\text{tr}}({a^2}) = 0$; the coefficients of $ {x^{n - 1}}$ and $ {x^{n - 2}}$ in the characteristic polynomial of $ a$ are thus 0. Also one gets a generic division algebra of degree $ n$ whose center has transcendence degree $ n + (n - 1)(n - 2)/2$, as well as a new (simpler) algebra of generic matrices. Equations are given to determine the cyclicity of these algebras, but they may not be tractable.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0732118-3
PII: S 0002-9947(1984)0732118-3
Article copyright: © Copyright 1984 American Mathematical Society