Specializations of Brauer Classes over Algebraic Function Fields

Let F be either a number field or a field finitely generated of transcendence degree ≥ 1 over a Hilbertian field of characteristic 0, let F (t) be the rational function field in one variable over F , and let α ∈ Br(F (t)). It is known that there exist infinitely many a ∈ F such that the specialization t→ a induces a specialization α→ α ∈ Br(F ), where α has exponent equal to that of α. Now let K be a finite extension of F (t) and let β = resK/F (t)(α). We give sufficient conditions on α and K for there to exist infinitely many a ∈ F such that the specialization t → a has an extension to K inducing a specialization β → β ∈ Br(K), K the residue field of K, where β has exponent equal to that of β. We also give examples to show that, in general, such a ∈ F