Indivisibility of divisor class numbers of Kummer extensions over the rational function field

We find a complete criterion for a Kummer extension K over the rational function field k = F-q(T) of degree l to have indivisibility of its divisor class number h(K) by l, where F-q is the finite field of order q and l is a prime divisor of q - 1. More importantly, when h(K) is not divisible by l, we have h(K) (math) 1 (mod l). In fact, the indivisibility of h(K) bye depends on the number of finite primes ramified in K/k and whether or not the infinite prime of k is unramified in K. Using this criterion, we explicitly construct an infinite family of the maximal real cyclotomic function fields whose divisor class numbers are divisible by l. (C) 2018 Elsevier Inc. All rights reserved.