Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T05:02:06.668Z Has data issue: false hasContentIssue false
  • English
  • Français

On p-Class Groups of Cyclic Extensions of Prime Degree p of Quadratic Fields

Published online by Cambridge University Press:  26 February 2010

Frank Gerth III
Frank Gerth III
Affiliation:
Department of Mathematics, University of Texas, Austin, Texas 78712, U.S.A.
Get access

Extract

Let Q denote the field of rational numbers, and let p be an odd prime number. Let K be a cyclic extension of Q of degree p, and let a be a generator of Gal (KQ). Let CK denote the p-class group of K (i.e., the Sylow p-subgroup of the ideal class group of K), and let for i = 1, 2, 3, . It is well known that is an elementary abelian p-group of rank tt1, where t is the number of ramified primes in KQ. So we focus our attention on . We let

Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 89 - 102
Copyright
Copyright University College London 1989

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 (Benjamin, New York, 1967).Google Scholar
2.Cassels, J. and Frhlich, A.. Algebraic Number Theory (Thompson Book Co., Washington, D.C., 1967).Google Scholar
3.Frohlich, A.. Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields (American Mathematical Society, Providence, R.I., 1983).Google Scholar
4.Gerth, F.. Counting certain number fields with prescribed l-class numbers. J. Reine Angew. Math., 337 (1982), 195207.Google Scholar
5.Gerth, F.. An application of matrices over finite fields to algebraic number theory. Math. Comp., 41 (1983), 229234.Google Scholar
6.Gerth, F.. Limit probabilities for coranks of matrices over GF((q). Linear and Multilinear Algebra, 19 (1986), 7993.CrossRefGoogle Scholar
7.Gerth, F.. Densities for ranks of certain parts of p-class groups. Proc. Amer. Math. Soc., 99 (1987), 18.Google Scholar
8.Gerth, F.. Densities for 3-class ranks in certain cubic extensions. J. Reine Angew. Math., 381 (1987), 161180.Google Scholar
9.Hardy, G. and , E.Wright. An Introduction to the Theory of Numbers, 4th edition (Oxford Univ. Press, London, 1965).Google Scholar
10.Lang, S.. Algebraic Number Theory (Addison-Wesley, Reading, Mass., 1970).Google Scholar