Geometry of 2 × 2 hermitian matrices.

Let D be a division ring which possesses an involution a → α . Assume that $$F = \{ a \in D|a = \overline a \} $$ is a proper subfield of D and is contained in the center of D. It is pointed out that if D 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 of F. 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 over D when n ≥ 3 and now can be deleted. When D is a field, the fundamental theorem of 2 × 2 hermitian matrices over D 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 [ABSTRACT FROM AUTHOR]