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The parity of the class number
of the cyclotomic fields of prime conductor

Author: Pietro Cornacchia
Journal: Proc. Amer. Math. Soc. 125 (1997), 3163-3168
MSC (1991): Primary 11R29, 11R18; Secondary 11R27
MathSciNet review: 1401730
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Abstract: Using a duality result for cyclotomic units proved by G.Gras, we derive a relation between the vanishing of some $\chi $-components of the ideal class groups of abelian fields of prime conductor (Theorem 1). As a consequence, we obtain a criterion for the parity of the class number of any abelian number field of prime conductor.

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  • 1. G. Cornell and M. I. Rosen, The $l$-rank of the real class group of cyclotomic fields, Compositio Math. 53 (1984), 133-141 MR 86d:11090
  • 2. A. Garbanati, Unit signatures, and even class numbers, and relative class numbers, Journal für die reine und angewandte Mathematik 274/275 (1975), 376-384
  • 3. A. Garbanati, Units with norm $-1$ and signatures of units, Journal für die reine und angewandte Mathematik 283/284 (1976), 164-75
  • 4. G. Gras and M.-N. Gras, Signatures des unités cyclotomiques et parité du nombre de classes des extensions cycliques de ${\mathbf {Q}}$ de degré premier impair, Ann. Inst. Fourier, Grenoble, 25, 1 (1975), 1-22 MR 52:13728
  • 5. G. Gras, Parité du nombre de classes et unités cyclotomiques, Astérisque 24-25 (1975), 37-45 MR 52:3109
  • 6. G. Gras, Critère de parité du nombre de classes des extensions abéliennes réeles de ${\mathbf {Q}}$ de degré impair, Bull. Soc. Math. France, 103 (1975), 177-190 MR 52:8081
  • 7. C. Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier, Grenoble 42, 3 (1992), 449-499 MR 93j:11071
  • 8. I. Hughes and R. Mollin, Totally positive units and squares, Proc. of the A.M.S., 87, 4 (1983), 613-616 MR 84d:12006
  • 9. F. Keqin, An elementary criterion on parity of class number of cyclic number field, Scientia Sinica (Series A), Vol.XXV (1982), 1032-1041
  • 10. S. Lang, Cyclotomic fields I and II, combined $2$nd edition, Graduate Texts in Math. 121, Springer Verlag, New York 1990 MR 91c:11001
  • 11. R. Schoof, The structure of the minus class groups of abelian number fields, Séminaire de Théorie des Nombres, Paris 1988-89, 185-204 MR 92e:11126
  • 12. R. Schoof, Minus class groups of the fields of the $l$-th roots of unity, to appear in Math. of Comp.
  • 13. W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980), 181-234 MR 82i:12004
  • 14. P. Stevenhagen, Class number parity for the $p$-th cyclotomic fields, Mathematics of Comp., Vol. 63, (1994), 773-784 MR 95a:11099
  • 15. L.C. Washington, Introduction to cyclotomic fields, Graduate Texts in Math. 83, Springer Verlag, New York 1982 MR 85g:11001

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Additional Information

Pietro Cornacchia
Affiliation: Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy

Received by editor(s): January 18, 1996
Received by editor(s) in revised form: May 17, 1996
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society

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