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Let k be a field of characteristic zero, let X be a geometrically integralk-variety of dimension n and let K be its field of fractions. Under theassumption that K contains all r-th roots of unity for an integer r, we provethat, given an element in H^m(K, Z/r), it becomes unramified in the extensionof K obtained by adding r-th roots of some n^2 functions in K.

A bound to kill the ramification over function fields