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Positive semidefinite forms over ordered skew fields

Author: Ka Hin Leung
Journal: Proc. Amer. Math. Soc. 106 (1989), 933-942
MSC: Primary 11E76; Secondary 11E81, 12J15, 16A39, 16A70, 16A86
MathSciNet review: 976367
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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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Article copyright: © Copyright 1989 American Mathematical Society

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