Simple cubic function fields and class number computations

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.