Serre's "formule de masse" in prime degree

TL;DR

This paper provides a detailed structural analysis of certain modules over local fields and offers an elementary proof of Serre's mass formula in degree p, along with explicit descriptions of related field extensions.

Contribution

It introduces a new method to compute contributions to Serre's mass formula and characterizes the compositum of all degree p extensions with solvable Galois closure over local fields.

Findings

Complete description of filtered modules over local fields.

Elementary proof of Serre's mass formula in degree p.

Explicit determination of the compositum of degree p extensions.

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^*) 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^*) or K(\wp^{-1}(K)) respectively, in the case of the local field F. Our method allows us to compute the contribution of each character G\to\F_p^* to the degree p mass formula, and, for any given group \Gamma, the contribution of those degree p separable extensions of F whose galoisian closure has group \Gamma.