Brauer-Hilbertian fields

Abstract

Let F F be a field of characteristic p p ( p = 0 p = 0 allowed), and let F ( t ) F(t) be the rational function field in one variable over F F . We say F F is Brauer-Hilbertian if the following holds. For every α \alpha in the Brauer group Br ⁡ ( F ( t ) ) \operatorname {Br}(F(t)) of exponent prime to p p , there are infinitely many specializations t → a ∈ F t \to a \in F such that the specialization α ¯ ∈ Br ⁡ ( F ) \bar \alpha \in \operatorname {Br}(F) is defined and has exponent equal to that of α \alpha . We show every global field is Brauer-Hilbertian, and if K K is Hilbertian and F F is finite separable over K ( t ) K(t) , F F is Brauer-Hilbertian.