Schmid's Formula for Higher Local Fields

TL;DR

This paper generalizes Schmid's formula to compute the Artin-Schreier-Witt-Parshin symbol for higher local fields, advancing understanding of ramification in two-dimensional local fields of positive characteristic.

Contribution

It extends Schmid's classical formula from one-dimensional local fields to two-dimensional local fields, providing a new tool for ramification analysis.

Findings

Derived explicit formula for Artin-Schreier-Witt-Parshin symbol

Applied the formula to two-dimensional local fields

Enhanced understanding of ramification in higher local fields

Abstract

In local class field theory, the Schmid-Witt symbol encodes interesting data about the ramification theory of $p$-extensi-ons of $K$ and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid-Witt symbol of Artin-Schreier extensions of local fields. Later, his formula was generalized to Artin-Schreier-Witt extensions, but still over a local field. In this paper we generalize Schmid's formula to compute the Artin-Schreier-Witt-Parshin symbol for Artin-Schreier-Witt extensions of two-dimensional local fields of positive characteristic.