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arXiv · 0906.3728

Function Fields with Class Number Indivisible by a Prime $\ell$

Michael Daub·Jaclyn Lang·Mona Merling·Allison M. Pacelli·Natee Pitiwan·Michael Rosen
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Abstract

It is known that infinitely many number fields and function fields of any degree $m$ have class number divisible by a given integer $n$. However, significantly less is known about the indivisibility of class numbers of such fields. While it's known that there exist infinitely many quadratic number fields with class number indivisible by a given prime, the fields are not constructed explicitly, and nothing appears to be known for higher degree extensions. In \cite{Pacelli-Rosen}, Pacelli and Rosen explicitly constructed an infinite class of function fields of any degree $m$, $3 \nmid m$, over $\F_q(T)$ with class number indivisible by 3, generalizing a result of Ichimura for quadratic extensions. Here we generalize that result, constructing, for an arbitrary prime $\ell$, and positive integer $m > 1$, infinitely many function fields of degree $m$ over the rational function field, with class number indivisible by $\ell$.

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Michael Daub, Jaclyn Lang, Mona Merling, Allison M. Pacelli, Natee Pitiwan, Michael Rosen. 2009-06-19. Function Fields with Class Number Indivisible by a Prime $\ell$. https://arxiv.org/abs/0906.3728

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