Classification of Hermitian forms and semisimple groups over fields of virtual cohomological dimension one (Q1911866)

The main objects of the reviewed paper are connected linear groups defined over a perfect field \(k\) with cohomological dimension of \(k(\sqrt {-1})\) at most one. A typical example of such a field is the field of rational functions \(\mathbb{R} (C)\) of a smooth projective real curve \(C\). The author gives a complete classification of semisimple groups over \(k\) thus generalizing classical results of \textit{E. Witt} [J. Reine Angew. Math. 171, 4-11 (1934; Zbl 0009.29103), 176, 31-44 (1937; Zbl 0015.05701)] on quadratic forms, and more recent results by \textit{N. Q. Thang} [Manuscr. Math. 78, No. 1, 9-35 (1993; Zbl 0804.11034); ibid. 82, No. 3-4, 445-447 (1994; Zbl 0812.11036)] on hermitian forms. The proof heavily depends on the results of another paper of the author [Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one (Preprint, Regensburg 1995)].