Summary: Journal of Pure and Applied Algebra 208 (2007) 833851
Projective Schur groups of Henselian fields
Eli Aljadeff, Jack Sonn, Adrian R. Wadsworth
Department of Mathematics, TechnionIsrael Institute of Technology, 32000 Haifa, Israel
Department of Mathematics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093-0112, USA
Received 24 June 2005; received in revised form 17 February 2006; accepted 3 March 2006
Available online 6 June 2006
Communicated by A.V. Geramita
One of the open questions that has emerged in the study of the projective Schur group PS(F) of a field F is whether or not
PS(F) is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that PS(F) = Br(L/F)?
We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no
counterexample is known. In this paper we prove that PS(F) is an algebraic relative Brauer group for all Henselian valued fields
F of equal characteristic whose residue field is a local or global field. For this, we first show how PS(F) is determined by PS(k)
for an equicharacteristic Henselian field with arbitrary residue field k.
c 2006 Elsevier B.V. All rights reserved.
MSC: 11R52; 11S25; 12F05; 12G05; 13A20
Let F be a field, and Br(F) its Brauer group. The Schur group S(F) of F is the subgroup of Br(F) consisting of