Final remarks on local discriminants

TL;DR

This paper demonstrates how to derive ramification filtrations in local number fields using Kummer theory and orthogonality relations, extending previous results to cases without primitive p-th roots of unity.

Contribution

It introduces a method to compute ramification filtrations and contributions to Serre's degree-p mass formula without requiring primitive p-th roots of unity.

Findings

Derived ramification filtration using Kummer theory and orthogonality.

Extended previous results to local fields lacking primitive p-th roots of unity.

Computed contributions of cyclic extensions to Serre's mass formula.

Abstract

We show how the ramification filtration on the maximal elementary abelian p-extension (p prime) on a local number field of residual characteristic p can be derived using only Kummer theory and a certain orthogonality relation for the Kummer pairing, even in the absence of a primitive p-th root of 1; the case of other local fields with finite residue fields was treated earlier. In all cases, we compute the contribution of cyclic extensions to Serre's degree-$p$ mass formula.