Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T20:54:44.044Z Has data issue: false hasContentIssue false
  • English
  • Français

Dihedral Field Extensions of Order 2p Whose Class Numbers are Multiples of p

Published online by Cambridge University Press:  20 November 2018

T. Callahan
T. Callahan*
Affiliation:
University of Toronto, Toronto, Ontario
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.

If L is a cyclic extension of Q of prime degree p, then the class number of L is divisible by p if and only if more than one prime divides the discriminant D, of L. If p ≠ 2, then this condition is equivalent to the existence of more than one cyclic extension of Q of degree p with discriminant equal to D. In this paper we generalize these results to non-galois extensions of Q of degree p whose normal closures have degree 2p over Q.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 2 , 01 April 1976 , pp. 429 - 439
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Angell, I. O., A table of complex cubic fields, Bull. London Math. Soc. 5 (1973), 37–38.Google Scholar
2. Artin, E. and Tate, J., Class field theory (Harvard, 1961).Google Scholar
3. Callahan, T., The 3-class groups of non-Galois cubic fields I, Mathematika 21 (1974), 72–89.Google Scholar
4. Callahan, T., The 3-class groups of non-Galois cubic fields II, Mathematika 21 (1974), 168–188.Google Scholar
5. Gorenstien, D., Finite groups (Harper and Row, 1968).Google Scholar
6. Hasse, H., Bericht ilber neuere Untersuchungen und Problem aus der Théorie der algraischen Zahlkôrper. Jahr. der D. Math. Ver., 35 (1926), 1-55; ibid. 36 (1927), 255-311; ibid. 39 (1930), 1–204.Google Scholar
7. Hasse, H., Arithmetiche Théorie der Rubischen Zahlkôrper auf Klassenkorpertheore-tischer Grundlage, Math. Zeit. 31 (1960), 565–582.Google Scholar
8. Heilbronn, H., Zeta functions and L-functions, in Algebraic number theory, edited by J., Cassels and Frôlich, A. (Thompson 1967).Google Scholar
9. Honda, T., Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7–12.Google Scholar
10. Martinet, J., Sur Varithmétique des extensions Galoisiennes à groupe de Galois diedral d'ordre 2 p. (Thèse, Grenoble 1968) Ann. Inst. Fourier, 19 (1969), 1–80.Google Scholar
11. Reichardt, H., Arithmetische théorie der rubischen kôrper als radikalkôrper, Monatshefte Math. Phys., 40 (1933), 323–350.Google Scholar