Genus Field and Extended Genus Field of an Elementary Abelian Extension of Global Fields

TL;DR

This paper constructs the genus and extended genus fields for elementary abelian l-extensions of rational function fields, utilizing Kummer theory and Dirichlet characters, especially when contained in cyclotomic fields.

Contribution

It provides explicit constructions of genus and extended genus fields for elementary abelian l-extensions, extending previous theories to function fields with cyclotomic considerations.

Findings

Explicit construction of genus fields for elementary abelian l-extensions.

Extension of the theory to include the extended genus field using Dirichlet characters.

Application of Leopoldt's ideas in the context of function fields.

Abstract

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian $l$-extension of a field of rational functions, where $l$ is a prime number. In the Kummer case, if $K$ is contained in a cyclotomic funtion field, the construction is given using Leopoldt's ideas by means of Dirichlet characters. Following the definition of Angl\`es and Jaulent of extended Hilbert class field, we obtain the extended genus field of an elementary abelian $l$-extension of a field of rational functions.