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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Class groups of imaginary function fields: The inert case


Authors: Yoonjin Lee and Allison M. Pacelli
Journal: Proc. Amer. Math. Soc. 133 (2005), 2883-2889
MSC (2000): Primary 11R29; Secondary 11R58
Published electronically: April 22, 2005
MathSciNet review: 2159765
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Abstract: Let $\mathbb{F} $ be a finite field and $T$ a transcendental element over $\mathbb{F} $. An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over $\mathbb{F} (T)$ in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers $m$ and $n$, there are infinitely many function fields $K$ of fixed degree $m$ such that the class group of $K$ contains a subgroup isomorphic to $(\mathbb{Z} /n\mathbb{Z} )^{m-1}$ and the prime at infinity is inert.


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Additional Information

Yoonjin Lee
Affiliation: Department of Mathematics, Smith College, Northampton, Massachusetts 01063
Email: yjlee@smith.edu

Allison M. Pacelli
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email: Allison.Pacelli@williams.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07910-4
PII: S 0002-9939(05)07910-4
Keywords: Class group, class number, rank of class group, imaginary function field
Received by editor(s): May 1, 2004
Received by editor(s) in revised form: June 8, 2004
Published electronically: April 22, 2005
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2005 American Mathematical Society