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Abstract:  Given a finite set of Brauer classes B of a fixed period \(ell\), we define ind(B) to be the g.c.d of degrees of field extensions L/F such that $alpha otimes _F L=0$ for every alpha in B. We provide upper-bounds for ind(B) which depends upon arithmetic invariants of fields of lower arithmetic complexity. As a simple application of our result, we will obtain upper-bounds for the splitting index of quadratic forms and finiteness of symbol length for function fields of curves over higher-local fields.