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

Cornacchia, Pietro

doi:10.1090/S0002-9939-97-03909-9

The parity of the class number of the cyclotomic fields of prime conductor
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by Pietro Cornacchia PDF

Proc. Amer. Math. Soc. 125 (1997), 3163-3168 Request permission

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.

References

Gary Cornell and Michael I. Rosen, The $\ell$-rank of the real class group of cyclotomic fields, Compositio Math. 53 (1984), no. 2, 133–141. MR 766293
A. Garbanati, Unit signatures, and even class numbers, and relative class numbers, Journal für die reine und angewandte Mathematik 274/275 (1975), 376–384
A. Garbanati, Units with norm $-1$ and signatures of units, Journal für die reine und angewandte Mathematik 283/284 (1976), 164–75
Georges Gras and Marie-Nicole Gras, Signature des unités cyclotomiques et parité du nombre de classes des extensions cycliques de $Q$ de degré premier impair, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 1, ix, 1–22 (French, with English summary). MR 392915
Georges Gras, Parité du nombre de classes et unités cyclotomiques, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) Astérisque, No. 24–25, Soc. Math. France, Paris, 1975, pp. 37–45 (French). D’après un travail en commun avec Marie-Nicole Gras. MR 0382224
Georges Gras, Critère de parité du nombre de classes des extensions abéliennes réelles de $Q$ de degré impair, Bull. Soc. Math. France 103 (1975), no. 2, 177–190 (French). MR 387238
Cornelius Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 449–499 (English, with English and French summaries). MR 1182638
I. Hughes and R. Mollin, Totally positive units and squares, Proc. Amer. Math. Soc. 87 (1983), no. 4, 613–616. MR 687627, DOI 10.1090/S0002-9939-1983-0687627-7
F. Keqin, An elementary criterion on parity of class number of cyclic number field, Scientia Sinica (Series A), Vol.XXV (1982), 1032–1041
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, DOI 10.1007/978-1-4612-0987-4
René Schoof, The structure of the minus class groups of abelian number fields, Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 185–204. MR 1104706
R. Schoof, Minus class groups of the fields of the $l$-th roots of unity, to appear in Math. of Comp.
W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980/81), no. 2, 181–234. MR 595586, DOI 10.1007/BF01389158
Peter Stevenhagen, Class number parity for the $p$th cyclotomic field, Math. Comp. 63 (1994), no. 208, 773–784. MR 1242060, DOI 10.1090/S0025-5718-1994-1242060-X
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2