The period-index conjecture for Brauer classes over $C_d$ fields

Let $K$ be a $C_d$ field, and let $\alpha\in \operatorname{Br}(K)$ be a Brauer class. The period $\operatorname{per}(\alpha)$ is its order in the Brauer group, and the index $\operatorname{ind}(\alpha)$ is the minimal degree of a field extension $L/K$ that splits $\alpha$. The period-index conjecture. For every $\alpha\in \operatorname{Br}(K)$, one has

$$\operatorname{ind}(\alpha)\mid \operatorname{per}(\alpha)^{d-1}.$$

This is a central question in the period–index problem, which asks how the index of a Brauer class is bounded in terms of its period. The conjecture has been established in numerous cases, although the general scope of the result depends on the class of $C_d$ fields under consideration.