N-Torsion of Brauer groups as relative Brauer groups of abelian extensions

Abstract

It is now known [H. Kisilevsky, J. Sonn, Abelian extensions of global fields with constant local degrees, Math. Res. Lett. 13 (4) (2006) 599–607; C.D. Popescu, Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields, J. Number Theory 115 (2005) 27–44] that if F is a global field, then the n-torsion subgroup Br n ( F ) of its Brauer group Br ( F ) equals the relative Brauer group Br ( L n / F ) of an abelian extension L n / F , for all n ∈ Z ⩾ 1 . We conjecture that this property characterizes the global fields within the class of infinite fields which are finitely generated over their prime fields. In the first part of this paper, we make a first step towards proving this conjecture. Namely, we show that if F is a non-global infinite field, which is finitely generated over its prime field and ℓ ≠ char ( F ) is a prime number such that μ ℓ 2 ⊆ F × , then there does not exist an abelian extension L / F such that Br ℓ ( F ) = Br ( L / F ) . The second and third parts of this paper are concerned with a close analysis of the link between the hypothesis μ ℓ 2 ⊆ F × and the existence of an abelian extension L / F such that Br ℓ ( F ) = Br ( L / F ) , in the case where F is a Henselian valued field.