Calculating formula and divisibility for relative class numbers of abelian function fields

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.