Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-10-02T08:20:17.589Z Has data issue: false hasContentIssue false
  • English
  • Français

A criterion for the parity of the class number of an Abelian field with prime power conductor

Published online by Cambridge University Press:  22 January 2016

Ken-Ichi Yoshino
Ken-Ichi Yoshino*
Affiliation:
Department of Mathematics, Kanazawa Medical University, Uchinada-machi, Ishikawa 920-02, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let f be a positive integer such that f ≢ 2 (mod 4). Let h0 be the class number of the maximal real subfield of fth cyclotomic field Q(ζf)- It is interesting to determine when h0 is even. Kummer [11] investigated this problem when f is a prime and showed that if h0 is even, then the relative class number h of the cyclotomic field is even (Satz III). Moreover he gave another necessary condition for h0 to be even (Satz IV). In [7] Hasse gave a necessary and sufficient condition for h to be even (Satz 45).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 145 , March 1997 , pp. 163 - 177
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1997

References

[ 1 ] Ankeny, N., Chowla, S. and Hasse, H., On the class number of the maximal real sub-field of a cyclotomic field, J. reine angew. Math., 217 (1965), 217–220.Google Scholar
[ 2 ] Borevich, Z.I. and Shafarevich, I. R., Number Theory, Academic Press, 1966.Google Scholar
[ 3 ] Cornell, G. and Rosen, M. I., The l-rank of the real class group of cyclotomic fields, Compositio Math., 53 (1984), 133–141.Google Scholar
[ 4 ] Fung, G., Granville, A. and Williams, H. C., Computation of the first factor of the class number of cyclotomic fields, J. Number Theory, 42 (1992), 297–312.CrossRefGoogle Scholar
[ 5 ] Gras, G., Critère de parité du nombre de classes des extensions abéliennes réelles de Q de degré impair, Bull. Soc. Math. France, 103 (1975), 177–190.CrossRefGoogle Scholar
[ 6 ] Gras, G. and Gras, M.-N., Signature des unités cyclotomiques et parité du nombre de classes des extensions cycliques de Q de degré premier impair, Ann. Inst. Fourier, Grenoble, 25 (1975), 1–22.Google Scholar
[ 7 ] Hasse, H., Uber die Klassenzahl abelscher Zahlkōrper, Akademie-Verlag, Berlin, 1952; Springer-Verlag, 1985.CrossRefGoogle Scholar
[ 8 ] Hazama, F., Demjanenko matrix, class number, and Hodge group, J. Number Theory, 34 (1990), 174–177.Google Scholar
[ 9 ] Hecke, E., Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea Pub. Co., 1948.Google Scholar
[10] Iwasawa, K., A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257–258.Google Scholar
[11] Kummer, E.E., Uber eine Eigenschaft der Einheiten der aus den Wurzeln der Gleichung a = 1 gebildeten complexen Zahlen, und über den zweiten Factor der Klassenzahl, Monatsber. Akad. Wiss., Berlin (1870), 855–880. Reprinted in Collected Papers, vol. I, Springer-Verlag, 1975, 919–944.Google Scholar
[12] Schwarz, W., Demjanenko matrix and 2-divisibility of class numbers, Arch. Math., 60(1993), 154–156.Google Scholar
[13] Washington, L.C., Introduction to cyclotomic fields, Springer-Verlag, 1982.Google Scholar
[14] Yoshino, K., On the class number of an abelian field with prime conductor, Proc. Japan Acad., 69A (1993), 278–281.Google Scholar