Communications of the
Korean Mathematical Society CKMS

In this paper, we construct explicitly an infinite family of primes $P$ with $h_P^{\pm} \equiv 0 \pmod {q^{\deg P}}$, where $h_P^{\pm}$ are the plus and minus parts of the divisor class number of the $P$-th cyclotomic function field over $\mathbb{F}_q(T)$. By using this result and Dirichlet's theorem, we give a condition of $A, M \in \mathbb{F}_q[T]$ such that there are infinitely many primes $P$ satisfying with $h_P^{\pm} \equiv 0 \pmod {p^e}$ and $P \equiv A \pmod M$.

Keywords: Class numbers, cyclotomic function fields

MSC numbers: Primary 11R29, 11R60