We shall mean by a number field a finite extension over the rational field Q contained in the complex field C, and by a CM-field a totally imaginary quadratic extension in C over a totally real number field. Let k be a totally real number field. Let Γ denote the set of all CM-fields that are quadratic extensions over k, so that Γ is an infinite set. In this paper, giving a characterization of CM-fields with odd relative class number, we shall prove that there exist infinitely many CM-fields in

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... with odd relative class number if and only if the class number of k in the narrow sense is odd. We shall also find out, by virtue of formulae of Kida [7], when Γ contains infinitely many CM-fields K such that µ − K