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.