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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).

References [Enhancements On Off] (What's this?)

  • [1] A. A. Albert, Structure of algebras, AMS Colloquium Series, vol. 24, 2nd ed., Amer. Math. Soc., Providence, R.I., 1961. MR 0123587 (23:A912)
  • [2] R. Brauer, On normal division algebras of index five, P.N.A.S., vol. 24 (1938), 243-246.
  • [3] J. H. M. Wedderburn, On division algebras, Trans. Amer. Math. Soc. 22 (1921), 129-135. MR 1501164

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