Specializations of Brauer classes over algebraic function fields

Let $F$ be either a number field or a field finitely generated of transcendence degree $\ge 1$ over a Hilbertian field of characteristic 0, let $F(t)$ be the rational function field in one variable over $F$, and let ${\alpha }\in \operatorname {Br}(F(t))$. It is known that there exist infinitely many $a\in F$ such that the specialization $t\to a$ induces a specialization ${\alpha }\to \overline {{\alpha }}\in \operatorname {Br}(F)$, where $\overline {{\alpha }}$ has exponent equal to that of ${\alpha }$. Now let $K$ be a finite extension of $F(t)$ and let ${\beta }=\operatorname {res}_{K/F(t)}({\alpha })$. We give sufficient conditions on ${\alpha }$ and $K$ for there to exist infinitely many $a\in F$ such that the specialization $t\to a$has an extension to $K$ inducing a specialization ${\beta }\to \overline {{\beta }}\in \operatorname {Br}(\overline{K})$, $\overline{K}$ the residue field of $K$, where $\overline {{\beta }}$ has exponent equal to that of ${\beta }$. We also give examples to show that, in general, such $a\in F$ need not exist.