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On the Class Number of a Relatively Cyclic Number Field

Published online by Cambridge University Press:  22 January 2016

Hideo Yokoi
Hideo Yokoi*
Affiliation:
Mathematical Institute, Nagoya University
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Let l be a rational prime. For each n≧0, denote by ζln a primitive ln-th root of unity and by Q(ζl,n) the cyclotomic field obtained by adjoining ζl,n to the rational field Q. Then a theorem which was proved by H. Weber is well known:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 29 , March 1967 , pp. 31 - 44
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1967

References

[1] Ankeny, N. C.-Chowla, S.-Hasse, H., Oa the class-number of the maximal real subfield of a cyclotomic field, J. reine angew. Math., 217 (1965), 217–220.Google Scholar
[2] Artin, E., Über Einheiten relativ Galoischer Zahlkorper, J. reine angew. Math., 167 (1931), 153–156.Google Scholar
[3] Brumer, A.-Rosen, M., Class number and ramification in number fields, Nägoya Math. J., 23 (1963), 97–101.Google Scholar
[4] Chevalley, C., Relation entre le nombre de classes d’un sous-corps et celui d’un surcorps, C. R. Sci. Paris, 192 (1931), 257–258.Google Scholar
[5] Chevaliey, C., Class field theory, (Th. 10.3), Notes at Nagoya University 1954.Google Scholar
[6] Furtwängler, Ph., Über die Klassenzahlen abelscher Zahlkörper, J. reine angew. Math., 134 (1908), 91–94.Google Scholar
[7] Furtwängler, Ph., Über die Klassenzahlen der Kreisteilungskörper, J. reine angew. Math., 140 (1911), 29–32.Google Scholar
[8] Furtwängler, Ph., Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper, Abh. Math. Sem. Hamburg, 7 (1930), 14–36.Google Scholar
[9] Hasse, H., Zur Geschlechtertheorie in quadratischen Zahlkörper, J. Math. Soc. Japan., 3 (1951), 45–51.Google Scholar
[10] Hasse, H., Bericht übef neuere Untersuchungen und Problème aus der Théorie der algebraischen Zahlkörper II, (1930), Jahresberichte der D.M.V.Google Scholar
[11] Honda, T., On absolute class fields of certain algebraic number field, J. reine angew. Math., 203 (1960). 80–89.CrossRefGoogle Scholar
[12] Iwasawa, K., A note on class numbers of algebraic number fields, Abh. Math. Sem. Hamburg, 20 (1956), 257–258.Google Scholar
[13] Iwasawa, K., A note on the group of units of algebraic number field, J. math, pure appl.. 35 (1956), 189–192.Google Scholar
[14] Iwasawa, K., A class number formula for cyclotomic fields, Ann. of Math., 76 (1962), 171–179.CrossRefGoogle Scholar
[15] Iyanaga, S.-Tamagawa, T., Sur la théorie du corps de classes sur le corps de nombres rationelles, J. Math. Soc. Japan, 3 (1951), 220–227.Google Scholar
[16] Kuroda, S.-N., Über die Klassenzahl eines relativzyklischen Zahlkörpers von Prim-zahlgrade, Proc. Japan Acad., 40 (1964), 623–626.Google Scholar
[17] Leopoldt, H. W., Zur Geschlechtertheorie in abelschen Zahlkörpern, Math. Nachr., 9 (1953), 351–362.Google Scholar
[18] Moriya, M., Über die Klassenzahl eines relativzyklischen Zahlkörpern von Primzahlgrad, Japanese J. Math., 10 (1933), 1–18.CrossRefGoogle Scholar
[19] Tannaka, T., Some remarks concerning principal ideal theorem, Töhoku Math. J., 1 (1949), 270–278.Google Scholar
[20] Terada, F., On a generalization of the principal ideal theorem, Töhoku Math. J., 1 (1949), 229–269.Google Scholar
[21] Weber, H., Théorie der algebraischen Zahlkörper, Acta Math., 8 (1886), 193–263.Google Scholar