BRAUER-HILBERTIAN FIELDS

Let F be a field of characteristic p (p = 0 allowed), and let F(t) be the rational function field in one variable over F . We say F is Brauer-Hilbertian if the following holds. For every alpha in the Brauer group Br(F(t)) of exponent prime to p , there are infinitely many specializations t --> a E F such that the specialization alphaBAR is-an-element-of Br(F) is defined and has exponent equal to that of alpha. We show every global field is Brauer-Hilbertian, and if K is Hilbertian and F is finite separable over K(t) , F is Brauer-Hilbertian.