Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-16T17:14:31.406Z Has data issue: false hasContentIssue false
  • English
  • Français

The 3-class groups of non-Galois cubic fields–II

Published online by Cambridge University Press:  26 February 2010

T. Callahan
T. Callahan
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada.
Get access

Extract

Let K3 be a non-Galois cubic extension of the rationals and let K6 be its normal closure. Under K6 there is a unique quadratic field K2. For i = 2, 3, 6 we define Cli to be the 3-class group of Ki and ri; to be the rank of Cli. In an earlier paper we examined the structure of Cl3 when K2 is complex and K6/K2 is unramified. In this paper we remove these restrictions and obtain similar results.

Type
Research Article
Information
Mathematika , Volume 21 , Issue 2 , December 1974 , pp. 168 - 188
Copyright
Copyright © University College London 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Artin, E. and Tate, J.. Class field theory (Harvard, 1961).Google Scholar
2.Barrucand, P. and Cohn, H.. “A rational genus, class number divisibility, and unit theory for pure cubic fields”, Journal of Number Theory, 2 (1970), 7–21.CrossRefGoogle Scholar
3.Barrucand, P. and Cohn, H.. “Remarks on principal factors in a relative cubic field”, Journal of Number Theory, 2 (1971), 226–239.CrossRefGoogle Scholar
4.Borel, A., Chowla, S., Herz, C. S., Iwasawa, K. and Serre, J.-P.. Seminar on complex multiplication (Springer-Verlag, Lecture Notes in Math. 21, 1966).CrossRefGoogle Scholar
5.Callahan, T., “The 3-class groups of non-Galois cubic fields”, Mathematika, 21 (1974), 72–89.CrossRefGoogle Scholar
6.Craig, M.. Irregular discriminants, Thesis (Ann Arbor, Mich., 1971).Google Scholar
7.Furtwängler, P.. “Beweis des Hauptidealsatz für die Klassenkörper algebraischer Zahlkörper”, Hamb. Sent. Abh., 7, 14–36.CrossRefGoogle Scholar
8.Gorenstien, D.. Finite groups (Harper and Row, 1968).Google Scholar
9.Gras, G.. Hamb. Sent. Abh., Thesis (Grenoble, France, 1972).Google Scholar
10.Hasse, H.. “Bericht über neuere Untersuchungen und Problem aus der Theorie der algebraischen Zahlkörper”, Jahr. der D. Math Ver., 35 (1926) 1–55; Jahr. der D. Math Ver., 36 (1927), 255–311; Jahr. der D. Math Ver., 39 (1930), 1–204.Google Scholar
11.Hasse, H.. “Arithmetische Theorie der kubischen Zahlkröper auf Klassenkörpertheoretischer Grundlage”, Math. Zeit., 31 (1930), 565–582.CrossRefGoogle Scholar
12.Honda, T.. “Pure cubic fields whose class numbers are multiples of three”, Journal of Number Theory, 3 (1971), 7–12.CrossRefGoogle Scholar
13.Reichardt, H.. “Arithmetische Theorie der kubischen Körper als Radikalkörper”, Monatshefte Math. Phys., 40 (1933), 323–350.CrossRefGoogle Scholar
14.Scholz, A.. “Idealklassen und Einheiten in kubischen Körpern”, Monatshefte Math. Phys., 40 (1933), 211–222.CrossRefGoogle Scholar
15.Scholz, A. and Taussky, O.. “Die Hauptideale der kubischen Klassenkörper imaginar-quadratischer Zahlkörper”, J. Reine Angew, Math., 171 (1934), 19–41.Google Scholar
16.Shanks, D.. “New types of quadratic fields having three invariants divisible by 3”, Journal of Number Theory, 4 (1972), 537–556.CrossRefGoogle Scholar
17.Shanks, D. and Serafin, R.. “Quadratic fields with four invariants divisible by 3”, Mathematics of Computation 27 (1973) 183–187.CrossRefGoogle Scholar
18.Shanks, D. and Weinberger, P.. “A quadratic field of prime discriminant requiring three generators for its class group, and related theory”, Acta Arithmetica, 21 (1972).CrossRefGoogle Scholar
19.Yoiko, H.. “On the class number of a relatively cyclic number field”, J. Math. Soc. Japan, 20 (1968), 411–418.Google Scholar