Ramification Groups of Nonabelian Kummer Extensions

The reciprocity law of Coleman for the Hilbert norm residue symbol has allowed the computation of the conductors of the abelian Kummer extensionsQ_p(a,ζ_pn)/Q_p(ζ _pn)witha∈Q_pandζ_pna 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 extensionQ_p(Q^×_p)/Q_pobtained from adjoining toQ_pallp-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.