  Let $F$ be the function field of a curve over a complete discretely valued
field. Let $\ell$ be a prime not equal to the characteristic of the residue
field. Given a finite subgroup $B$ in the $\ell$ torsion part of the Brauer
group ${}_{\ell}Br(F)$, we define the index of $B$ as the minimum of the
degrees of field extensions which split all elements in $B$. In this
manuscript, we give an upper bound for the index of any finite subgroup $B$ in
terms of arithmetic invariants of $F$. As a simple application of our result,
given a quadratic form $q/F$, where $F$ is the function field of a curve over
an $n$-local field, we provide an upper bound to the minimum of degrees of
field extensions $L/F$ so that the Witt index of $q\otimes L$ becomes the
largest possible.
