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≡1(modℓ). In fact, the indivisibility of h_K by ℓ 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.