Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



$p$-class groups of certain extensions of degree $p$

Author: Christian Wittmann
Journal: Math. Comp. 74 (2005), 937-947
MSC (2000): Primary 11R29, 11R33, 11Y40
Published electronically: October 27, 2004
MathSciNet review: 2114656
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $p$ be an odd prime number. In this article we study the distribution of $p$-class groups of cyclic number fields of degree $p$, and of cyclic extensions of degree $p$ of an imaginary quadratic field whose class number is coprime to $p$. We formulate a heuristic principle predicting the distribution of the $p$-class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to-$p$-part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.

References [Enhancements On Off] (What's this?)

  • 1. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
  • 2. Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313
  • 3. H. COHEN AND H.W. LENSTRA, Heuristics on class groups of number fields, Number Theory Noordwijkerhout 1983, LNM 1068, Springer, 1984.MR 0756082 (85j:11144)
  • 4. Henri Cohen and Jacques Martinet, Étude heuristique des groupes de classes des corps de nombres, J. Reine Angew. Math. 404 (1990), 39–76 (French). MR 1037430
  • 5. S.D. FISHER AND M.N. ALEXANDER, Matrices over a finite field, Amer. Math. Monthly 73 (1966), 639-641.
  • 6. Georges Gras, Sur les 𝑙-classes d’idéaux dans les extensions cycliques relatives de degré premier 𝑙. I, II, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 1–48; ibid. 23 (1973), no. 4, 1–44 (French, with English summary). MR 0360519
  • 7. F. GERTH III, Counting certain number fields with prescribed $l$-class numbers, J. Reine Angew. Math. 337 (1982), 195-207.MR 0676052 (84c:12002)
  • 8. F. GERTH III, Densities for ranks of certain parts of $p$-class groups, Proc. Amer. Math. Soc. 99 (1987), 1-8. MR 0866419 (88b:11067)
  • 9. Frank Gerth III, On 𝑝-class groups of cyclic extensions of prime degree 𝑝 of quadratic fields, Mathematika 36 (1989), no. 1, 89–102. MR 1014203,
  • 10. Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028
  • 11. Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859
  • 12. L. R´EDEI, Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper, J. Reine Angew. Math. 171 (1935), 55-60.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R29, 11R33, 11Y40

Retrieve articles in all journals with MSC (2000): 11R29, 11R33, 11Y40

Additional Information

Christian Wittmann
Affiliation: Universität der Bundeswehr München, Fakultät für Informatik, Institut für Theoretische Informatik und Mathematik, 85577 Neubiberg, Germany

Keywords: Class groups, Galois modules, Cohen-Lenstra heuristics, numerical verifications
Received by editor(s): March 21, 2003
Received by editor(s) in revised form: March 27, 2004
Published electronically: October 27, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society