Indivisibility of class numbers of global function fields

Function fields with class number indivisible by a prime l

“…In 1999, Ichimura [12] constructed an explicit infinite family of quadratic function fields with class number not divisible by 3. Pacelli and Rosen [32] extended this to non-quadratic fields of degree m over F q (T ), 3 ∤ m. In this paper, we generalize Pacelli and Rosen's result, constructing, for a large class of q, infinitely many function fields of any degree m over F q (T ) with class number indivisible by an arbitrary prime ℓ.…”

“…As in [32], the fields we construct are given explicitly. The idea of the proof is to construct two towers of fields N 1 ⊂ · · · ⊂ N t = F q (T ) and M 1 ⊂ · · · ⊂ M t .…”

“…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 [32], Pacelli and Rosen explicitly constructed an infinite class of function fields of any degree m, 3 ∤ 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 ℓ, and positive integer m > 1, infinitely many function fields of degree m over the rational function field, with class number indivisible by ℓ.…”

Simple cubic function fields and class number computations

“…In this paper, our goal is to study the function field (over a finite field F q ) analogue of these polynomials. While several authors have considered the immediate analogue of Shanks' polynomial, including [10,23], the lack of a detailed study of these cubic function fields is the primary motivation for this work. In the function field setting, one might ask: "are there other starting points?"…”