The aim of this paper is to clarify the ordinarity of cyclotomic function fields. In the previous work [J. Number Theory 133 (2013)], the author determined all monic irreducible polynomials m such that the maximal real subfield of the mth cyclotomic function field is ordinary. In this paper, we extend this result to the general case.

Let $K={𝔽}_q\left(T\right)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f\left(T\right)\in {𝔽}_q\left[T\right]$ with positive degree, denote by ${\Lambda }_f$ the torsion points of the Carlitz module for the polynomial ring ${𝔽}_q\left[T\right]$. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K\left({\Lambda }_P\right)$ of degree $k$ over ${𝔽}_q\left(T\right)$, where $P\in {𝔽}_q\left[T\right]$ is an irreducible polynomial of positive degree and $k>1$ is a positive divisor of $q-1$. A formula for the analytic class number for the...