  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 $\ell$.
