ISRAEL JOURNAL OF MATHEMATICS 131 (2002), 375-379 FIELDS FOR WHICH THE PROJECTIVE SCHUR SUBGROUP IS THE WHOLE BRAUER GROUP BY ELI ALJADEFF AND JACK SONN Department of Mathematics, Technion -- Israel Institute of Technology Haifa 32000, Israel e-mail: aljadeff@math.technion.ac.il, sonn@math.technion.ac.il ABSTRACT Let Br(K) denote the Brauer group of a field K and PS(K) the projective Sehur subgroup. i. Let K be a finitely generated infinite field. Then PS(K) = Br(K) if and only if I( is a global field. 2. Let /x ~ be a finitely generated infinite field, and let K((t)) denote the field of formal power series in t over /iv. Then PS(K((t))) --- Br(K((t))) if and only if K = Q. 1. Introduction Let K be any field. The projective Schur (sub)group PS(K) of a field K is the subgroup of the Brauer group Br(K) generated by (in fact, consisting of) all classes that are represented by a projective Schur algebra A. A finite dimensional K-central simple algebra A is a projective Schur algebra over K if the group of units A* of A contains a subgroup F which spans A as a k-vector space and is finite modulo the center, i.e., K'F//(* is a finite group. The notions of projective Schur algebra and the projective Schur group are the projective analogues of Schur algebra and the Schur group of K. These analogues were introduced in 1978 by Lorenz and Opolka [10]. A symbol algebra is a projective Schur algebra in an obvious way (indeed, let A = (a, b)n be the symbol algebra generated by x and y subject to the relations x ~ = a c K*, y~ = b ~ K*, yx = ~xy, where ~n E K* Received November 4, 2001 375