Ramification Groups of Nonabelian Kummer Extensions

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Abstract

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)/Qppn)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.

Original languageEnglish (US)
Pages (from-to)105-115
Number of pages11
JournalJournal of Number Theory
Volume65
Issue number1
DOIs
StatePublished - Jul 1 1997

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ASJC Scopus subject areas

  • Algebra and Number Theory

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