Brauer-Hilbertian fields

Fein, Burton; Saltman, David J.; Schacher, Murray

doi:10.1090/S0002-9947-1992-1075382-0

Brauer-Hilbertian fields
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by Burton Fein, David J. Saltman and Murray Schacher PDF

Trans. Amer. Math. Soc. 334 (1992), 915-928 Request permission

Abstract:

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 $\operatorname {Br}(F(t))$ of exponent prime to $p$, there are infinitely many specializations $t \to a \in F$ such that the specialization $\bar \alpha \in \operatorname {Br}(F)$ 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.

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