Torsion Subgroups of Brauer Groups and Extensions of Constant Local Degree for Global Function Fields

We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non–p–part of the Picard group Pic0(k) of k, there exists an abelian extension L/k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above, the n–torsion subgroup Brn(k) of the Brauer group Br(k) of k is algebraic, answering a question of Aldjaeff and Sonn in this context.