Class fields, Dirichlet characters and extended genus fields of global function fields

TL;DR

This paper characterizes the extended genus field of abelian extensions of rational function fields using class field theory and Dirichlet characters, establishing equivalence with existing definitions and exploring their properties.

Contribution

It demonstrates the equivalence of different definitions of extended genus fields for cyclotomic function fields and extends the study to general abelian extensions using class field theory.

Findings

The natural definition of extended genus field matches Anglès and Jaulent's definition for cyclotomic function fields.

The extended genus field of abelian extensions can be characterized via Dirichlet characters.

Comparison of approaches enhances understanding of genus fields in function field theory.

Abstract

We obtain the extended genus field of an abelian extension of a rational function field. We follow the definition of Angl\`es and Jaulent, which uses class field theory. First we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Angl\`es and Jaulent. Next we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields and compare this approach with the one given by Angl\`es and Jaulent.