The reciprocity law of Coleman for the Hilbert norm residue symbol has allowed the computation of the conductors of the abelian Kummer extensionsQp(a,ζpn)/Qp(ζ pn)witha∈Qpandζpna 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 extensionQp(Q×p)/Qpobtained 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.
ASJC Scopus subject areas
- Algebra and Number Theory