A note on the relative class number in function fields

Let $F$ be a finite field, $A=F[T]$, and $k=F(T)$. Let $K_{m}=k(\Lambda _{m})$ be the field extension of $k$ obtained by adjoining the $m$-torsion on the Carlitz module. The class number $h_{m}$ of $K_{m}$ can be written as a product $h_{m}=h_{m}^{+}h_{m}^{-}$. The number $h_{m}^{-}$ is called the relative class number. In this paper a formula for $h_{m}^{-}$ is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of $p$-th roots of unity. Some consequences of this formula are also derived.