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The 2-Ideal Class Groups of ℚ(ζl)

Published online by Cambridge University Press:  22 January 2016

Pietro Cornacchia
Pietro Cornacchia*
Affiliation:
Corso XXV Aprile 60, 14100 Asti, Italy, [email protected]
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Abstract

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For prime l we study the structure of the 2-part of the ideal class group Cl of ℚ(ζl). We prove that Cl ⊗ ℤ2) is a cyclic Galois module for all l < 10000 with one exception and compute the explicit structure in several cases.

Keywords

11R2911R1811R27
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 162 , June 2001 , pp. 1 - 18
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

References

[1] Berthier, T., Générateurs et structure du groupe des classes d’idéaux des corps denombres abéliens, thése, Univ. de Franche-Comté, 1994.Google Scholar
[2] Conner, P. E. and Hurrelbrink, J., Class number parity, Series in pure mathematics 8, World Scientific, Singapore 1988.Google Scholar
[3] Cornacchia, P., The parity of the class number of the cyclotomic fields of prime con ductor, Proceedings of the A.M.S., 125, n.11 (1997), 3163–3168.Google Scholar
[4] Cornacchia, P., Anderson’s module for cyclotomic fields of prime conductor, J. Number The ory, 67, 2 (1997), 252–276.Google Scholar
[5] Cornacchia, P. and Greither, C., Fitting Ideals of Class Groups of Real Fields with Prime Power Conductor, J. Number Theory, 73 (1998), 459–471.CrossRefGoogle Scholar
[6] Gras, G., Nombre de φ-classes invariantes, application aux classes des corps abéliens, Bull. Soc. Math. de France, 106 (1978), 337–364.Google Scholar
[7] Gras, G. and Berthier, T., Sur la structure des groupes de classes relatives, avec un appendice d’exemples numériques, Ann. Inst. Fourier, Grenoble, 43, 1 (1993), 1–20.Google Scholar
[8] Gras, M. N., Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q, J. Reine Angew. Math., 277 (1975), 89–116.Google Scholar
[9] Greither, C., Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier, Grenoble, 42, 3 (1992), 449–499.Google Scholar
[10] Guerry, G., Sur la 2-composante du groupe des classes de certaines extensions cycliques de degré 2N , J. Number Theory, 53 (1995), 159–172.CrossRefGoogle Scholar
[11] Lang, S., Cyclotomic fields I and II, combined 2nd edition, Graduate Texts in Math., 121, Springer Verlag, New York 1990.Google Scholar
[12] Oriat, B., Relation entre les 2-groupes des classes d’idéaux au sens ordinaire et restreint de certains corps de nombres, Bull. Soc. math. France, 104 (1976), 301–307.CrossRefGoogle Scholar
[13] Rubin, K., Global units and ideal class groups, Invent. Math., 89 (1987), 511–526.Google Scholar
[14] Rubin, K., The Main Conjecture, Appendix to [11].Google Scholar
[15] Schoof, R., Minus class groups of the fields of the l-th roots of unity, Math. of Comp., 67 (1998), 1225–1245.Google Scholar
[16] Schoof, R., The structure of the minus class groups of abelian number fields, Séminaire de Théorie des Nombres, Paris 1988–89, 185–204.Google Scholar
[17] Schoof, R., Class numbers of real cyclotomic fields of prime conductor, preprint.Google Scholar
[18] Washington, L.C., Introduction to cyclotomic fields, 2nd edition, Graduate Texts in Math. 83, Springer Verlag, New York 1997.Google Scholar