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We extend the "character sum method" for the computation of densities in Artin primitive root problems given by Lenstra and the authors  to the situation of radical extensions of arbitrary rank. It leads to dramatic shortenings of the existing higher rank proofs, and enables us to extend their range of application in a more or less straightforward way.doi:10.4064/aa163-1-2 fatcat:27fyqw3yfnai7c3njdu4k6vi2e