Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field ${\bb Q}$ , are contained in the complex field ${\bb C}$ . Let $P$ be the set of all prime numbers. For any algebraic number field $F$ , let $C_F$ denote the ideal class group of $F$ and, writing $F^+$ for the maximal real subfield of $F$ , let $C^-_F$ denote the kernel of the norm map from $C_F$ to the ideal class group of $F^+$ ; for each $l \in P$ , let $C_F(l)$ denote the $l$ -class group of $F$ , that is, the $l$ -primary component of $C_F$ , and let $C^-_F(l)$ denote the $l$ -primary component of $C^-_F$ . Furthermore, for each $l \in P$ , we denote by ${\bb Z}_l$ the ring of $l$ -adic integers.