  We give a formula for the essential dimension of a cohomology class $\alpha$
in $H^d(K, \mathbb{Q}_p/\mathbb{Z}_p (d))$ 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.
