Let \(F\) be a CM-field; in its cyclotomic tower \(\cup_{n}F_ n\), Iwasawa’s formula for the \(p\)-relative class number of \(F_ n\) is characterized by the classical \(\mu^-_ F\), \(\lambda^-_ F\) invariants. If \(E/F\) is a Galois extension of CM-fields, of \(p\)-power degree, the author proves, in full generality, that \(\mu^-_ E=0\Leftrightarrow \mu^-_ F=0,\) and that, when \(\mu^-_ F=0\), \(\lambda^-_ E\) is given explicitly, from \(\lambda^-_ F\), via Y. Kida’s formula [J. Number Theory 12, 519–528 (1980; Zbl 0455.12007)].
The proof of the author involves only the theory of \(p\)-adic \(L\)-functions of totally real fields, and uses the properties of the corresponding \(p\)-adic pseudomeasures of Deligne-Ribet; the main argument is that if \(\chi\),\(\psi\) are even characters, \(\psi\) of \(p\)-power order, the congruence “ \(\chi \psi \equiv \chi mod \pi\) ” (for an evident \(\pi \mid p\)) gives a congruence between \(L^*_ p(\chi \psi)\) and the product of \(L^*_ p(\chi)\) with suitable Euler factors (this has been observed also by K. A. Ribet [Sémin. Delange-Pisot-Poitou, 19e Année 1977/78, Théor. des Nombres, Fasc. 1, Exp. 9 (1978; Zbl 0394.12007)]), where the \(L^*_ p\) are suitable series giving \(L_ p\)-functions of characters of the cyclotomic tower; this gives a relation between analytic \(\lambda\)-invariants \(\lambda\) (\(\chi \psi)\), \(\lambda\) (\(\chi)\), and then Kida’s formula comes from the classical analytic class number formula involving \(L\)-functions at \(s=0.\) As it is explained by the author, the case of abelian extensions \(F,E\) over \(\mathbb Q\) was given by the reviewer, and the Galois representation aspects of Kida’s theory, by K. Iwasawa [Tôhoku Math. J., II. Ser. 33, 263–288 (1981; Zbl 0468.12004)].