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A Note on Ideal Class Groups

Published online by Cambridge University Press:  22 January 2016

Kenkichi Iwasawa
Kenkichi Iwasawa*
Affiliation:
Massachusetts Institute of Technology
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In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 239 - 247
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

1) N. Tschebotarew, Zur Gruppentheorie des Klassenkörpers, J reine u. angew. Math., 161 (1929), pp. 179-183.

2) For the classical class field theory used here and in the following, see H. Hasse’s “Klassenkörperbericht” in Jahresbericht D. M.-V., 1926, 1927, 1930.

3) K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg, 20 (1956), pp. 257-258.

4) For a more complete result in this direction, see H. Hasse, Über die Klassenzahl abelscher Zahlkòrper, Berlin, 1952, § 37.

5) Iwasawa, op. cit.

6) Hasse, Klassenzahl, p. 150.

7) Hasse, Klassenzahl, p. 152.

8) E. Kummer, Über die Irregularität der Determinanten, Monatsber. Akad. d. Wissensch., Berlin, 1853, pp. 194-200.

9) Hasse, Klassenzahl, p. 99.

10) K. Iwasawa, A class number formula for cyclotomic fields, Ann, of Math., 76 (1962), pp. 171-179,

11) Kummer, op. cit.