Class numbers of some abelian extensions of rational function fields

Let $P$ be a monic irreducible polynomial. In this paper we generalize the determinant formula for $h(K_{P^n}^+)$ of Bae and Kang and the formula for $h^{-}(K_{P^n})$ of Jung and Ahn to any subfields $K$ of the cyclotomic function field $K_{P^n}.$ By using these formulas, we calculate the class numbers $h^{-}(K), h(K^+)$ of all subfields $K$ of $K_P$ when $q$ and $\deg(P)$ are small.