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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 indoi:10.4064/aa-67-3-219-227 fatcat:3podcecyffbj7mhi7xp3irqp2i