Math. Ann. 297, 575-580 (1993) kalalm 9 Springer-Verlag 1993 A question on the discriminants of involutions of central division algebras R. Parimala, R. Sridharan, V. Suresh School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India Received 17 November 1992 Mathematics Subject Classification (1991): 16K20, 12G05 Introduction Let k be a field of characteristic not equal to 2 and D a finite dimensional central simple algebra over k whose class is 2-torsion in the Brauer group Br(k) of k. By classical results of Albert [A, Ch. X, Th. 19], D admits an involution which is identity on k. An invariant with values in k* modulo squares in k* called dis- criminant (rather pfaffian discriminant) was associated to an involution on D in [KPS1] and it was shown in [KPS2] that this invariant precisely gives the obstruction to the splitting of an involution on a rank 16 algebra into a tensor product of involutions on quaternion subalgebras. Since then, this invariant has been an interesting object of investigation. If D is not a division algebra, it is an easy exercise that the set of discriminants of orthogonal involutions on D coincides with the group of reduced norms from D* modulo squares in k*. Let D be a central division algebra over k. If the degree of D over k is 2, then every orthogonal involution on D is of the form Int(u)o z0, ~0 being the "canonical" involution on D and u an element of D such that z0(u) = -u. Thus the set of discriminants of orthogonal involutions on D is the set of square classes of reduced norms of trace zero elements of D. This certainly is not a group in general and may not contain the element 1. Let D be any central division algebra of degree ~ 4 over k, which is 2-torsion in Br(k). The following basic questions concerning discriminants of involutions on D were raised by several mathematicians including Knus, Lam, Rowen, Saltman, Tignol and Yanchevskii. (Q) Does D admit an orthogonal involution of discriminant 17 (Qs) Is the set of discriminants of orthogonal involutions on D equal to the full group of reduced norms of non-zero elements of D modulo squares in k* ? Obviously an affirmative answer to (Qs) gives an affirmative answer to (Q). Since every reduced norm from D* is a product of two discriminants, (Qs) is equivalent to the question whether the set of discriminants of orthogonal involutions is a group.