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Prime factors of cyclotomic class numbers


Author: D. H. Lehmer
Journal: Math. Comp. 31 (1977), 599-607
MSC: Primary 12A50; Secondary 12A35
DOI: https://doi.org/10.1090/S0025-5718-1977-0432589-6
MathSciNet review: 0432589
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Abstract: Let p be an odd prime. The "first factor" $ {h^\ast}(p)$ of the class number of the field of pth roots of unity has been the subject of many investigations beginning with Kummer (1861). In the present paper it is shown how the theory of a function introduced by T. A. Pierce (1917) can be used to find the prime factors of $ {h^\ast}(p)$.


References [Enhancements On Off] (What's this?)

  • [1] E. KUMMER, "Bestimmung der Anzahl nicht äquivalenter Classen für die aus $ \lambda $ ten Wurzeln der Einheit gebildeten complexen Zahlen und die idealen Factoren derselben," J. Reine Angew. Math., v. 40, 1850, pp. 43-116 (p. 110, formula 38).
  • [2] D. H. LEHMER, "Factorization of certain cyclotomic functions," Ann. of Math. (2) v. 34, 1933, pp. 461-479 (p. 463, Theorem 3). MR 1503118
  • [3] Z.I. BOREVICH (BOREVIČ) & I. R. SHAFAREVICH (ŠAFAREVIČ), Number Theory, "Nauka", Moscow, 1964; English transl., Pure and Appl. Math., vol. 20, Academic Press, New York, 1966. MR 30 #1080; 33 #4001. MR 0195803 (33:4001)
  • [4] V. A. LEBESGUE, "Dimostrazione dell'irreduttibilitá dell'equazione formata con le radici primitive dell'unitá," Ann. Mat. Pura Appl., v. 2, 1859, pp. 232-237.
  • [5] W. J. LEVEQUE, Topics in Number Theory, Vol. 1, Addison-Wesley, Reading, Mass., 1956, p. 48 (Theorem 4.1). MR 18, 283. MR 0080682 (18:283d)
  • [6] T. METSÄNKYLÄ, "On prime factors of the relative class numbers of cyclotomic fields," Ann. Univ. Turku. Ser. A I, No. 149, 1971, 8 pp. MR 44 #178. MR 0282944 (44:178)
  • [7] G. SCHRUTKA V. RECHTENSTAMM, "Tabelle der (Relativ)-Klassenzahlen der Kreiskörper, deren $ \phi $-Funktion des Wurzelexponenten (Grad) nicht grösser als 256 ist," Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech., 1964, no. 2, 64 pp. MR 29 #4918. MR 0167646 (29:4918)
  • [8] M. NEWMAN, "A table of the first factor for prime cyclotomic fields," Math. Comp., v. 24, 1970, pp. 215-219. MR 41 #1684. MR 0257029 (41:1684)
  • [9] D. H. LEHMER & J. M. MASLEY, "Table of the cyclotomic class numbers $ {h^\ast}(p)$ and their factors for $ 200 < p < 512$." (To appear.)

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DOI: https://doi.org/10.1090/S0025-5718-1977-0432589-6
Article copyright: © Copyright 1977 American Mathematical Society

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