Genus fields of Kummer extensions of rational function fields

TL;DR

This paper determines the genus field of general Kummer extensions over rational function fields, providing a comprehensive understanding of their algebraic structure and extending previous results to more general cases.

Contribution

It introduces a method to compute the genus field of any Kummer extension of a rational function field, generalizing known results for prime power degrees.

Findings

The genus field of a composite of two abelian extensions equals the composite of their genus fields.

The main result explicitly describes the genus of a general Kummer extension.

The approach simplifies the calculation of genus fields for complex extensions.

Abstract

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.