Genus fields of Kummer $\ell^n$-cyclic extensions

TL;DR

This paper constructs the genus field for Kummer $ ext{ell}^n$-cyclic extensions over rational function fields, generalizing previous results and analyzing related extension properties.

Contribution

It provides a new construction method for genus fields of Kummer cyclic extensions and extends existing results to more general cases.

Findings

Computed genus fields within cyclotomic function fields.

Generalized Peng's results to $ ext{ell}^n$-cyclic extensions.

Analyzed extension properties of combined genus fields.

Abstract

We give a construction of the genus field for Kummer $\ell^n$-cyclic extensions of rational congruence function fields, where $\ell$ is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer $\ell$-cyclic extension. Finally, we study the extension $(K_1K_2)_{\frak{ge}}/(K_1)_{\frak{ge}}(K_2)_{\frak{ge}}$, for $K_1$, $K_2$ abelian extensions of $k$.