Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields
For a field \(k\) and a natural number \(m\) denote by \(\text{Br}_m(k)\) the \(m\)-torsion subgroup of the Brauer group Br\((k)\) of \(k\). In their paper ``Relative Brauer groups and \(m\)-torsion'', Proc. Am. Math. Soc. 130, No. 5, 1333--1337 (2002; Zbl 1099.11066) \textit{E. Aljadeff} and \textit{J. Sonn} raised the question whether Br\(_m(k)\) is algebraic, i.e., equal to the kernel \(\text{Br}(L/k)\) of the restriction homomorphism \(\text{Br}(k)\to \text{Br}(L)\) for some algebraic separable extension \(L/k\), and examples were given there such that \(\text{Br}_m(k)\) is not algebraic. In the paper under review the author considers this question in the case where \(k\) is a global field of characteristic \(p\), i.e., \(k\) is a finite extension of the rational function field in one variable over the field with \(p\) elements, and obtains the following result in this case: If \(L/k\) is a Galois extension of local degree \(m\) at every place of \(k\) then \(\text{Br}_m(k)=\text{Br}(L/k)\). Motivated by this result the author investigates the existence of Galois extensions \(L/k\) such that their local degrees at every place of \(k\) are equal to a given natural number \(m\). One of his results in this respect is as follows: There is an abelian Galois extension \(L/k\) with this property provided \(m\) is coprime to the order of the non-\(p\)-part of the Picard group of \(k\).
scientific article; zbMATH DE number 5009834
scientific article