# Genus fields of global fields

## TL;DR

This paper characterizes the extended genus field of global fields, providing explicit descriptions for both function fields and number fields using class field theory techniques.

## Contribution

It introduces the concept of extended genus fields for global fields and derives their descriptions via class field theory, extending previous understanding.

## Key findings

- Explicit description of extended genus fields for function fields.
- Application of function field techniques to number fields.
- Use of class field theory to characterize these fields.

## Abstract

In this paper we obtain the extended genus field of a global field. First we define the extended genus field of a global function field and we obtain, via class field theory, the description of the extended genus field of an arbitrary global function field. In the last part of the paper we use the techniques for function fields to describe the extended genus field of an arbitrary number field.