  We provide a uniform bound for the index of cohomology classes in $H^i(F,
\mu_\ell^{\otimes i-1})$ when $F$ is a semiglobal field (i.e., a one-variable
function field over a complete discretely valued field $K$). The bound is given
in terms of the analogous data for the residue field of $K$ and its finitely
generated extensions of transcendence degree at most one. We also obtain
analogous bounds for collections of cohomology classes. Our results provide
recursive formulas for function fields over higher rank complete discretely
valued fields, and explicit bounds in some cases when the information on the
residue field is known. In the process, we prove a splitting result for
cohomology classes of degree 3 in the context of surfaces over finite fields.
