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.