On the projective Schur group of a field
The authors disprove a conjecture of \textit{P. Nelis} and \textit{F. Van Oystaeyen} [J. Algebra 137, 501-518 (1991; Zbl 0735.13003)] by which the projective Schur group \(PS(k)\) of a field \(k\), as has been defined by \textit{F. Lorenz} and \textit{H. Opolka} [Math. Z. 162, 175-182 (1978; Zbl 0384.16009)], should coincide with the whole Brauer group \(Br(k)\) of \(k\). In the counter-example \(k\) is a rational function field in one variable over an infinite field that is finitely generated over its prime field. Over such fields there exist cyclic algebras of which no splitting field is contained in the composite of a cyclotomic extension and a Kummer extension of \(k\); a projective Schur algebra, however, is shown to admit an abelian splitting field that is a subfield of a radical extension of \(k\).
rational function fields in one variable
scientific article; zbMATH DE number 840716