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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A useful proposition for division algebras of small degree


Author: Darrell Haile
Journal: Proc. Amer. Math. Soc. 106 (1989), 317-319
MSC: Primary 16A39; Secondary 12E15
MathSciNet review: 972232
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Abstract: Let $ F$ be a field and let $ D$ be an $ F$-central division algebra of degree $ n$. We present a short, elementary proof of the following statement: There is an $ n - 1$-dimensional $ F$-subspace $ V$ of $ D$ such that for every nonzero element $ \nu $ of $ V$, $ {\text{Tr}}(\nu ) = {\text{Tr(}}{\nu ^{ - 1}}) = 0$. We then indicate how one can use this result to obtain the basic structural results on division algebras of degree three and four (results of Wedderburn and Albert, respectively).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0972232-0
PII: S 0002-9939(1989)0972232-0
Article copyright: © Copyright 1989 American Mathematical Society