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arXiv · 1108.6048

Simple cubic function fields and class number computations

Pieter Rozenhart·Jonathan Webster
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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.

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Pieter Rozenhart, Jonathan Webster. 2011-08-30. Simple cubic function fields and class number computations. https://arxiv.org/abs/1108.6048

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