Projective Schur groups of Henselian fields

Abstract

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 .