Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the ideal class groups of imaginary abelian fields with small conductor


Authors: Kuniaki Horie and Hiroko Ogura
Journal: Trans. Amer. Math. Soc. 347 (1995), 2517-2532
MSC: Primary 11R29; Secondary 11R20
DOI: https://doi.org/10.1090/S0002-9947-1995-1297529-6
MathSciNet review: 1297529
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ k$ be any imaginary abelian field with conductor not exceeding 100, where an abelian field means a finite abelian extension over the rational field $ {\mathbf{Q}}$ contained in the complex field. Let $ C(k)$ denote the ideal class group of $ k$, $ {C^ - }(k)$ the kernel of the norm map from $ C(k)$ to the ideal class group of the maximal real subfield of $ k$, and $ f(k)$ the conductor of $ k;f(k) \leqslant 100$. Proving a preliminary result on $ 2$-ranks of ideal class groups of certain imaginary abelian fields, this paper determines the structure of the abelian group $ {C^ - }(k)$ and, under the condition that either $ [k:{\mathbf{Q}}] \leqslant 23$ or $ f(k)$ is not a prime $ \geqslant 71$, determines the structure of $ C(k)$.


References [Enhancements On Off] (What's this?)

  • [1] H. Bauer, Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper, J. Number Theory 1 (1969), 161-162. MR 0240072 (39:1426)
  • [2] G. Cornell and M. Rosen, Group-theoretic constraints on the structure of the class group, J. Number Theory 13 (1981), 1-11. MR 602445 (82e:12005)
  • [3] F. Gerth, The ideal class groups of two cyclotomic fields, Proc. Amer. Math. Soc. 78 (1980), 321-322. MR 553367 (80k:12004)
  • [4] H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952, and Springer-Verlag, Berlin, 1985.
  • [5] K. Horie, On the exponents of ideal class groups of cyclotomic fields, Proc. Amer. Math. Soc. 119 (1993), 1049-1052. MR 1169030 (94a:11166)
  • [6] K. Horie and M. Horie, On the $ 2$-class groups of cyclotomic fields whose maximal real subfields have odd class numbers, Proc. Amer. Math. Soc. (to appear). MR 1169030 (94a:11166)
  • [7] K. Iwasawa, A note on ideal class groups, Nagoya Math. J. 27 (1966), 239-247. MR 0197438 (33:5603)
  • [8] Y. Kida, $ l$-extensions of CM-fields and cyclotomic invariants, J. Number Theory 12 (1980), 519-528. MR 599821 (82c:12006)
  • [9] E. E. Kummer, Über die Irregularität von Determinanten, Monatsber. Akad. Wiss. Berlin 1853, 194-200; reprinted in his Collected Papers. Vol. I, pp. 539-545.
  • [10] -, Über die Klassenzahl der aus $ n$-ten Einheitswurzeln gebildeten complexen Zahlen, Monatsber. Akad. Wiss. Berlin 1861, 1051-1053; reprinted in his Collected Papers. Vol. I, pp. 883-885.
  • [11] F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), 693-707. MR 669662 (84e:12005)
  • [12] J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), 297-319. MR 511747 (80e:12005)
  • [13] G. Schrutka von Rechtenstamm, Tabelle der Relativ-Klassenzahlen der Kreiskörper, deren $ \phi $-Funktion des Wurzelexponenten (Grad) nicht grösser als 256 ist, Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. 1964, no. 2. MR 0167646 (29:4918)
  • [14] W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980), 181-234. MR 595586 (82i:12004)
  • [15] K. Tateyama, On the ideal class groups of some cyclotomic fields, Proc. Japan Acad. Ser. A 58 (1982), 333-335. MR 682697 (84b:12011)
  • [16] L. C. Washington, Introduction to cyclotomic fields, Springer-Verlag, Berlin, 1982. MR 718674 (85g:11001)
  • [17] K. Yoshino and M. Hirabayashi, On the relative class number of the imaginary abelian number field. I, Mem. Coll. Liberal Arts Kanazawa Medical Univ. 9 (1981), 5-53.
  • [18] -, On the relative class number of the imaginary abelian number field. II, Mem. Coll. Liberal Arts Kanazawa Medical Univ. 10 (1982), 33-81.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11R29, 11R20

Retrieve articles in all journals with MSC: 11R29, 11R20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1297529-6
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society