Bull. Austral. Math. Soc. 72(3) pp.471--476, 2005.

On 3-class groups of certain pure cubic fields

Frank Gerth III

Received: 23rd August, 2005



Recently Calegari and Emerton made a conjecture about the 3-class groups of certain pure cubic fields and their normal closures. This paper proves their conjecture and provides additional insight into the structure of the 3-class groups of pure cubic fields and their normal closures.

Click to download PDF of this article (free access until July 2006)

or get the no-frills version

[an error occurred while processing this directive]
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199648 Z'blatt-MATH: pre05031546


  1. P. Barrucand and H. Cohn;
    Remarks on principal factors in a relative cubic field,
    J. Number Theory 3 (1971), pp. 226--239. MR276197
  2. P. Barrucand, H. Williams and L. Baniuk;
    A computational technique for determining the class number of a pure cubic field,
    Math. Comp. 30 (1976), pp. 312--323. MR392913
  3. F. Calegari and M. Emerton;
    On the ramification of Hecke algebras at Eisenstein primes,
    Invent. Math. 160 (2005), pp. 97--144. MR2129709
  4. F. Gerth;
    On 3-class groups of pure cubic fields,
    J. Reine Angew. Math. 278/279 (1975), pp. 52--62. MR387234
  5. F. Gerth;
    Ranks of 3-class groups of non-Galois cubic fields,
    Acta Arith. 30 (1976), pp. 307--322. MR422198
  6. G. Gras;
    Sur les -classes d'idéaux des extensions non galoisiennes de {Q} de degré premier impair a clôture galoisienne diédrale de degré 2ℓ,
    J. Math. Soc. Japan 26 (1974), pp. 677--685. MR364179
  7. T. Honda;
    Pure cubic fields whose class numbers are multiples of three,
    J. Number Theory 3 (1971), pp. 7--12. MR292795
  8. L. Washington;
    Introduction to cyclotonic fields (Springer-Verlag, New York, 1982). MR718674

ISSN 0004-9727