Abstract. We prove that there are exactly six normal CM-fields of degree 32 with class number one. Five of them are composita of two normal CM-fields of degree 16 with the same maximal totally real octic field.

The educational approach of empathy has been prominently mentioned as a possible solution to the dehumanization occurring as a byproduct of the rapid development of science and technology in modern society. This study aims to identify the characteristics of Korean adolescents on the empathy factor, as a core competency of twenty-first century education, and the applicability of the empathy scale. This study analyzed 688 Korean adolescents' cognitive and affective empathy using the Interpersonal Reactivity Index (IRI). Korean adolescents showed an increase in all four factors of IRI as they got older. According to gender, females' empathy was higher in all four factors. In terms of the validity of the scale, the difference in subscales by gender was obvious. But, the instability in subscales by age of the Perfective Taking (PT) and Personal Distress (PD) was shown. Especially, PD had an insignificant correlation with other subscales. By revealing the limitations of the IRI, this study suggested ways to improve it. Therefore, this study will contribute to the educational foundation of Korean adolescents' empathic ability.

In this study, we attempted to investigate the empathy ability of Korean adolescents. Davis' Interpersonal Reactivity Index was applied to 1155 high school students in Korea, and the results were compared with those obtained from the survey of the American and Dutch students with similar ages. As a result, high school students in Korea had higher cognitive empathy than other countries. But, in cognitive empathy, Fantasy (FN) was lower than Perspective Taking (PT), unlike other countries. The mean score of women in all empathy subscales was higher than that of men. And there is significant difference between humanities and natural science high school students in PT, Empathic Concern (EC) and Personal Distress (PD). However, in the comparison of mean score's effect size and correlation in the subscales of the empathetic scale, it was confirmed as an unstable structure in PD. Therefore, there is a need to educationally discuss how to improve this limitation of measuring empathy in the further research.

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In this paper we determine all non-normal quartic CM-fields with relative class number two and all octic dihedral CM-fields with relative class number two: there are exactly 254 non-isomorphic non-normal quartic CM-fields with relative class number two and 95 non-isomorphic octic dihedral CM-fields with relative class number two. This result permits us to find all octic dihedral CM-fields of which ideal class groups are nontrivial and cyclic of 2-power order: there are precisely 16 non-isomorphic such octic dihedral CM-fields. Academic Press