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.

The first part of the proof of Lemma 3 given in the author's paper [1] is obviously incomplete. Namely, [Kfr ]v(ζ) in the fifth line from the bottom of page 245 of [1] is not necessarily a cyclic extension over [Kfr ]v−1(ζ) in the next line. To complete the proof, we shall prove an assertion after simple preliminaries, as follows. Let p be an odd prime. For any CM-field k of finite or infinite degree, let k+ denote the maximal real subfield of k, A(k) the p-class group of k (that is, the p-primary component of the ideal class group of k), A(k+) the p-class group of k+, and A(k)− the kernel of the norm map A(k)→A(k+). Since p>2, A(k)−={a∈A(k)[mid ]aj=a−1} where j denotes the complex conjugation of k. For any extension F′/F of CM-fields, let ϕF′/F denote the homomorphism A(F)−→A(F′)− induced by the natural map A(F)→A(F′). Now, the following completes the proof in question.