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 extensions Q(p)((pn)root alpha, zeta(pn))/Q(p)(zeta(pn)) with alpha is an element of Q(p) and zeta(pn) a primitive (p(n))th root of unity for a fixed prime p and all positive integers n. From these conductors, we compute the ramification,groups of the nonabelian Kummer extension Q(p)((p proportional to)root)Q(p)(x))/Q(p) obtained from adjoining to Q(p) all p-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. (C) 1997 Academic Press.