Essential dimension of cohomology classes via valuation theory

We give a formula for the essential dimension of a cohomology class $\alpha$ in $H^d(K,{\mathbb{Q}}_p\mathrm{/}{\mathbb{Z}}_p(d))$ when $K$ is a strictly Henselian field. This formula is particularly explicit in the case, where $\alpha$ is a Brauer class (for $d=2$). As an application of our bound with $d=3$, we study the essential dimension of exceptional groups by examining the image of the Rost invariant.