Abstract
LetD be a division ring which possesses an involution a → α . Assume that\(F = \{ a \in D|a = \overline a \} \) is a proper subfield ofD and is contained in the center ofD. It is pointed out that ifD is of characteristic not two, D is either a separable quadratic extension of F or a division ring of generalized quaternions over F and that if D is of characteristic two,D is a separable quadratic extension ofF. Thus the trace map Tr:D → F, a → a + a is always surjective, which is formerly posed as an assumption in the fundamental theorem of n×n hermitian matrices overD when n ≥ 3 and now can be deleted. WhenD is a field, the fundamental theorem of 2 × 2 hermitian matrices overD has already been proved. This paper proves the fundamental theorem of 2×2 hermitian matrices over any division ring of generalized quaternions of characteristic not two
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This research was completed during a visit to the Academy of Mathematics and System Sciences, Chinese Academy of Sciences.
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Huang, L., Wan, Z. Geometry of 2 × 2 hermitian matrices. Sci. China Ser. A-Math. 45, 1025–1037 (2002). https://doi.org/10.1007/BF02879986
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DOI: https://doi.org/10.1007/BF02879986