Ramification Groups of Nonabelian Kummer Extensions Original Research Article

The reciprocity law of Coleman for the Hilbert norm residue symbol has allowed the computation of the conductors of the abelian Kummer extensionsimagewithaset membership, variantQpandζpna primitive (pn)th root of unity for a fixed primepand all positive integersn. From these conductors, we compute the ramification groups of the nonabelian Kummer extensionimageobtained from adjoining toQpallp-power roots of its elements. More generally, given a similar nonabelian Kummer extension of complete discrete valuation fields, we have a method of computing its ramification groups from the conductors of the abelian Kummer extensions and knowledge of the ramification groups of the cyclotomic extensions.