Simple cubic function fields and class number computations

Details Simple cubic function fields and class number computations Simple cubic function fields and class number computations

2011-08-30

arxiv.org/abs/1108.6048v2

Mathematics

Number Theory

Typos fixed, extra references added, extra information on error bounds for class number approximations, timings and algorithm

Scientific paper

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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