A note on a refined formula of the relative class number of an imaginary abelian field

We deduce the formula (1) in [4] for an imaginary abelian field K from Theorem 3 of A. Wiles [3] which holds for a certain abelian CM-extension of a totally real algebraic number field. Since the maximal abelian p-extension F contained in K is totally real when pis odd, the formula follows from (1) from this result in [4] by applying it to F under the 邸sumption that the Iw邸awa polynomials do not have non-trivial zeros.