Simple cubic function fields and class number computations
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 o...
Saved in:
Main Authors | , |
---|---|
Format | Journal Article |
Language | English |
Published |
30.08.2011
|
Subjects | |
Online Access | Get full text |
DOI | 10.48550/arxiv.1108.6048 |
Cover
Summary: | 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. |
---|---|
DOI: | 10.48550/arxiv.1108.6048 |