On the $2$-class groups of cyclotomic fields whose maximal real subfields have odd class numbers
HTML articles powered by AMS MathViewer
- by Kuniaki Horie and Mitsuko Horie PDF
- Proc. Amer. Math. Soc. 123 (1995), 2643-2649 Request permission
Abstract:
This paper first makes a simple remark about the structure as a group of the ideal class group of any algebraic number field, then studies the 2-rank of the ideal class group of any cyclotomic field whose maximal real subfield has odd class number, and finally determines the structure of the ideal class groups of some cyclotomic fields.References
- Gary Cornell and Michael I. Rosen, The $\ell$-rank of the real class group of cyclotomic fields, Compositio Math. 53 (1984), no. 2, 133–141. MR 766293 H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie Verlag, Berlin, 1952; Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1985.
- Kuniaki Horie, On the exponents of ideal class groups of cyclotomic fields, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1049–1052. MR 1169030, DOI 10.1090/S0002-9939-1993-1169030-5
- Kenkichi Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258. MR 83013, DOI 10.1007/BF03374563 E. E. Kummer, Über die Klassenzahl der aus n-ten Einheitswurzeln gebildeten complexen Zahlen, Monatsber. Akad. Wiss. Berlin (1861), 1051-1053; Collected papers, I, pp. 883-885.
- F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693–707. MR 669662, DOI 10.1090/S0025-5718-1982-0669662-5
- John Myron Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), no. 3, 297–319. MR 511747
- J. Myron Masley and Hugh L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286(287) (1976), 248–256. MR 429824
- Guntram Schrutka von Rechtenstamm, Tabelle der (Relativ)-Klassenzahlen der Kreiskörper, deren $\varphi$-Funktion des Wurzelexponenten (Grad) nicht grösser als 256 ist, Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 1964 (1964), no. 2, 64 (German). MR 167646
- W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math. (2) 108 (1978), no. 1, 107–134. MR 485778, DOI 10.2307/1970932
- Koichi Tateyama, On the ideal class groups of some cyclotomic fields, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 7, 333–335. MR 682697
- Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2 K. Yoshino and M. Hirabayashi, On the relative class number of the imaginary abelian number field I, Mem. College Liberal Arts Kanazawa Medical Univ. 9 (1981), 5-53.
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2643-2649
- MSC: Primary 11R18; Secondary 11R29
- DOI: https://doi.org/10.1090/S0002-9939-1995-1273498-5
- MathSciNet review: 1273498