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Class number computations of real abelian number fields

Author: F. J. van der Linden
Journal: Math. Comp. 39 (1982), 693-707
MSC: Primary 12A50; Secondary 12A35
MathSciNet review: 669662
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Abstract: In this paper we describe the calculation of the class numbers of most real abelian number fields of conductor $ \leqslant 200$. The technique is due to J. M. Masley and makes use of discriminant bounds of A. M. Odlyzko. In several cases we have to assume the generalized Riemann hypothesis.

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  • [1] Senon I. Borewicz and Igor R. Šafarevič, Zahlentheorie, Aus dem Russischen übersetzt von Helmut Koch. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 32, Birkhäuser Verlag, Basel-Stuttgart, 1966 (German). MR 0195802
  • [2] Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
  • [3] A. Fröhlich, On the class group of relatively Abelian fields, Quart. J. Math. Oxford Ser. (2) 3 (1952), 98–106. MR 52459,
  • [4] 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
  • [5] Marie-Nicole Gras, 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 (French). MR 389845,
  • [6] M. N. Gras, "Table numérique de nombre de classes et des unités des extensions cycliques réelles de degré 4 de Q," Publ. Math. Fac. Sci. Besançon, 1978.
  • [7] Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (German). MR 0049239
  • [8] The theory of numbers (1975), xi+541. With contributions by T. Tannaka, T. Tamagawa, I. Satake, Akira Hattori, G. Fujisaki and H. Shimizu; Translated from the Japanese by K. Iyanaga; North-Holland Mathematical Library, Vol. 8. MR 0444603
  • [9] Serge Lang, Cyclotomic fields, Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 59. MR 0485768
  • [10] H. W. Leopoldt, Über Einheitengruppe und Klassenzahl reeller abelscher Zahlkörper, Abh. Deutsch. Akad. Wiss. Berlin. Kl. Math. Nat. 1953 (1953), no. 2, 48 pp. (1954) (German). MR 0067927
  • [11] Sirpa Mäki, The determination of units in real cyclic sextic fields, Lecture Notes in Mathematics, vol. 797, Springer, Berlin, 1980. MR 584794
  • [12] Jacques Martinet, Tours de corps de classes et estimations de discriminants, Invent. Math. 44 (1978), no. 1, 65–73 (French). MR 460281,
  • [13] John Myron Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), no. 3, 297–319. MR 511747
  • [14] A. M. Odlyzko, Discriminant Bounds, Unpublished tables (Nov. 1976).
  • [15] G. Schrutka von Rechtenstamm, Tabelle der (Relativ)-Klassenzahlen der Kreiskörper, Akademic-Verlag, Berlin, 1964.
  • [16] J.-P. Serre, Représentations Linéaries des Groups Finis, Hermann, Paris, 1967.
  • [17] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math. (2) 108 (1978), no. 1, 107–134. MR 485778,
  • [18] 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,
  • [19] Samuel S. Wagstaff Jr., The irregular primes to 125000, Math. Comp. 32 (1978), no. 142, 583–591. MR 491465,
  • [20] N. G. de Bruijn, On the factorization of cyclic groups, Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 (1953), 370–377. MR 0059271
  • [21] F. Diaz y Diaz, "Tables minorant la racine n-ième du discriminant d'un corps de degré n," Publ. Math. Orsay 80, 1980.
  • [22] Georges Poitou, Minorations de discriminants (d’après A. M. Odlyzko), Séminaire Bourbaki, Vol. 1975/76 28ème année, Exp. No. 479, Springer, Berlin, 1977, pp. 136–153. Lecture Notes in Math., Vol. 567. MR 0435033
  • [23] Georges Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année: (1976/77), Théorie des nombres, Fasc. 1 (French), Secrétariat Math., Paris, 1977, pp. Exp. No. 6, 18 (French). MR 551335

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Keywords: Class number, cyclotomic field, abelian number field
Article copyright: © Copyright 1982 American Mathematical Society