Radical Extensions for the Carlitz--Hayes Module

TL;DR

This paper investigates radical extensions generated by roots of Carlitz-Hayes polynomials over function fields, focusing on radical cyclotomic extensions and their properties related to torsion and extension order.

Contribution

It characterizes radical cyclotomic extensions, proving they have order a power of the base field's characteristic and providing bounds for their Carlitz-Hayes torsion.

Findings

Radical cyclotomic extensions have order a power of the characteristic.

Bounds are established for the Carlitz-Hayes torsion in these extensions.

The study advances understanding of the structure of radical extensions in function fields.

Abstract

Let $L/K$ be a finite extension of congruence function fields. We say that $L/K$ is a {\it radical extension} if $L$ is generated by roots of polynomials $u^{M}-\alpha \in K[u]$, where $u^{M}$ is the action of Carlitz-Hayes. We study a special class of these extensions, the {\it radical cyclotomic} extensions. We prove that any radical cyclotomic extension has order a power of the characteristic of $K$. We also give bounds for the Carlitz-Hayes torsion of these extensions.