Genus fields of finite abelian extensions

TL;DR

This paper determines the genus field of finite abelian extensions over the rational function field, introduces the conductor of constants, and provides explicit descriptions for p-extensions.

Contribution

It introduces the conductor of constants concept and explicitly describes the genus field for finite abelian p-extensions, advancing understanding of their structure.

Findings

Explicit formulas for the genus field of finite abelian extensions

Introduction of the conductor of constants concept

Detailed description of genus fields for p-extensions

Abstract

In this paper we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the particular case of finite abelian $p$--extensions and give an explicit description of their genus field.