Abstract
For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F p[G]-module K × /K × p in characteristic 0 and \({K/\wp(K)}\) in characteristic p, where \({K=F(\root{p-1}\of{F^\times})}\) and G = Gal(K|F). As an application, we give an elementary proof of Serre’s mass formula in degree p. We also determine the compositum C of all degree-p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is \({K(\root p\of{K^\times})}\) or \({K(\wp^{-1}(K))}\), respectively, in the case of the local field F.
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Communicated by Umberto Zannier.
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Dalawat, C.S. Serre’s formule de masse in prime degree. Monatsh Math 166, 73–92 (2012). https://doi.org/10.1007/s00605-010-0274-0
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DOI: https://doi.org/10.1007/s00605-010-0274-0