Simple cubic function fields and class number computations

TL;DR

This paper investigates simple cubic function fields, introduces the concept of $k$-exceptional units, and computes class numbers and invariants, including class number one fields, over finite fields.

Contribution

It provides a new proof characterizing Galois simple cubic function fields and computes their class numbers and invariants, extending the understanding of cubic function fields.

Findings

Galois simple cubic function fields are analogous to Shanks' simplest cubic number fields.

Class numbers of these fields over $ ext{F}_5$ and $ ext{F}_7$ are computed using truncated Euler products.

All Galois simple cubic function fields with class number one are determined under mild restrictions.

Abstract

In this paper, we study simple cubic fields in the function field setting, and also generalize the notion of a set of exceptional units to cubic function fields, namely the notion of $k$-exceptional units. We give a simple proof that the Galois simple cubic function fields are the immediate analog of Shanks simplest cubic number fields. In addition to computing the invariants, including a formula for the regulator, we compute the class numbers of the Galois simple cubic function fields over $\mathbb{F}_{5}$ and $\mathbb{F}_{7}$ using truncated Euler products. Finally, as an additional application, we determine all Galois simple cubic function fields with class number one, subject to a mild restriction.