CORRECTION TO ‘TWO ASPECTS OF THE RELATIVE k-INVARIANT’

Abstract

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 ] a^j=a⁻¹} where j denotes the complex conjugation of k. For any extension F′/F of CM-fields, let ϕ_F′/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.