Artin-Hasse formula for $p^m-$primary elements

TL;DR

This paper derives a new form of the Artin-Hasse formula for p-primary elements using Borevich's generators, connecting it with unramified p-extensions and Lubin-Tate formal groups.

Contribution

It provides a novel derivation of the Artin-Hasse formula and extends it to Lubin-Tate formal groups under specific unramified extension conditions.

Findings

Derived the Artin-Hasse formula from Borevich's generators

Expressed the Hilbert symbol in terms of generator expansions for Lubin-Tate groups

Connected p-th roots and unramified p-extensions in the formula

Abstract

Using Borevich's system of generators and relations, the classical Artin-Hasse formula is obtained from scratch in the case when taking $p$-th root of the second argument of the Hilbert symbol gives an unramified $p$-extension of the same degree of irregularity. Under the same assumptions, in the case of Lubin-Tate formal groups, an expression for the Hilbert symbol is obtained in terms of the expansion of elements by the same system of generators.