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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Positive semidefinite forms over ordered skew fields
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by Ka Hin Leung PDF
Proc. Amer. Math. Soc. 106 (1989), 933-942 Request permission

Abstract:

In any ordered field $K$, the inequality ${x^2} + {y^2} \geq 2xy$ obviously holds for any $x,y \in K$. Naturally, we may ask if the same inequality holds in every ordered skew field. Surprisingly, it can be proved that in an ordered domain $R$, the above inequality holds for any elements $x,y$ in $R$ iff $R$ is commutative. In this paper, we formulate a generalization of the above observation and prove that if a "positive semidefinite form" over an ordered skew field admits a "nontrivial" solution, then the skew field is actually a field.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 106 (1989), 933-942
  • MSC: Primary 11E76; Secondary 11E81, 12J15, 16A39, 16A70, 16A86
  • DOI: https://doi.org/10.1090/S0002-9939-1989-0976367-8
  • MathSciNet review: 976367