2016/08/15 by Chandan Singh Dalawat, Dalawat, Chandan Singh
Mathematics · #11S15 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Primary 11R23
paper · pdf · doi:10.48550/arxiv.1608.04182
openalex publication_date 2016/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be a prime number, let K be a p-field (a local field with finite residue field of characteristic p), let L be a finite galoisian tamely ramified extension of K, and let G=Gal(L|K). Suppose that L is split over K in the sense that the short exact sequence 1→ T→ G→ G/T→1 has a section, where T is the inertia subgroup of G. We determine the structure of the Fp[G]-module L^× /L× p in characteristic 0 when the p-torsion subgroup pL^× of L^× has order p, and of the Fp[G]-modules L^× /L× p and L+ /\wp(L+) in characteristic p, where \wp(x)=xp-x. Let K be a maximal galoisian extension of K, let V be the maximal tamely ramified extension of K in K, let Γ=Gal(V|K), and let B be the maximal abelian extension of exponent p of V in K. We determine the structure of the Fp[[Γ]]-module Gal(B|V), and show how this leads in characteristic 0 to a simple proof of the fact that the profinite group Gal( K|K) is generated by [K:Qp]+3 elements.