This is a shortened and slightly modified English version of our paper "Sur une classe des corps de nombres algébriques et ses applications", published in this journal in 1975; see Gyory . In that paper we studied an important class of number fields, namely the totally imaginary quadratic extensions of totally real number fields. We obtained among others some new norm inequalities and discriminant inequalities in such number fields. That time several di erent names were used in the literature for these number fields. This is the reason that the title of Gyory  is not informative enough. Probably it is partly due to this fact that the results of  are less known. Nowadays, the name CM-field is generally accepted for the number fields under consideration. The purpose of this paper is to better call the attention to our norm inequalities and discriminant inequalities in CM-fields and to their applications.
- Discriminant of algebraic number
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