A complete classification of cubic function fields over any finite field

TL;DR

This paper provides a comprehensive classification of cubic function fields over finite fields, developing a Galois theory that accounts for missing roots of unity and offering explicit criteria for irreducibility, ramification, and Galois actions.

Contribution

It introduces a complete Galois theory for cubic function fields over finite fields, including cases with missing roots of unity, and provides explicit criteria and descriptions for their structure.

Findings

Complete classification of cubic function fields over finite fields.

Explicit criteria for irreducibility and ramification.

Descriptions of Galois actions and integral bases.

Abstract

We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow one to easily read ramification and splitting data from the generating equation, in analogy to the known theory for Artin-Schreier and Kummer extensions. We also describe explicit irreducibility criteria, integral bases, and Galois actions in terms of canonical generating equations.