Class number formulae in the form of a product of determinants in function fields

Abstract

In this paper, we generalize the Koera's group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an inter-mediate field in the tower of cyclotomic function fields with prime power conductor.