# Further remarks on local discriminants

## TL;DR

This paper develops methods using Kummer and Artin-Schreier theories to compute ramification filtrations and discriminants of elementary abelian p-extensions over local fields, extending classical results.

## Contribution

It introduces a unified approach to analyze elementary abelian p-extensions over local fields using Kummer and Artin-Schreier theories, providing explicit computations.

## Key findings

- Computed ramification filtrations for elementary abelian p-extensions
- Derived discriminant formulas for these extensions
- Extended Artin-Schreier theory to finite extensions of (())

## Abstract

Using Kummer theory for a finite extension K of \Qp(\zeta)(where p is a prime number and \zeta a primitive p-th root of~1), we compute the ramification filtration and the discriminant of an arbitrary elementary abelian p-extension of K. We also develop the analogous Artin-Schreier theory for finite extensions of \Fp((\pi)) and derive similar results for their elementary abelian p-extensions.