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

TL;DR

This paper constructs infinitely many function fields of any degree greater than one over the rational function field with class numbers not divisible by a given prime, extending previous results for specific cases.

Contribution

It generalizes prior work by explicitly constructing infinitely many function fields of arbitrary degree with class number indivisible by any prime.

Findings

Constructed infinite families of function fields with class number indivisible by a prime.

Extended previous quadratic case results to arbitrary degrees.

Provided explicit constructions over the rational function field.

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 $\ell$.