Prime factors of cyclotomic class numbers
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- by D. H. Lehmer PDF
- Math. Comp. 31 (1977), 599-607 Request permission
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
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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).
- D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461–479. MR 1503118, DOI 10.2307/1968172
- A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803 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.
- William Judson LeVeque, Topics in number theory. Vols. 1 and 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682
- Tauno Metsänkylä, On prime factors of the relative class numbers of cyclotomic fields, Ann. Univ. Turku. Ser. A I 149 (1971), 8. MR 282944
- Guntram Schrutka von Rechtenstamm, Tabelle der (Relativ)-Klassenzahlen der Kreiskörper, deren $\varphi$-Funktion des Wurzelexponenten (Grad) nicht grösser als 256 ist, Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 1964 (1964), no. 2, 64 (German). MR 167646
- Morris Newman, A table of the first factor for prime cyclotomic fields, Math. Comp. 24 (1970), 215–219. MR 257029, DOI 10.1090/S0025-5718-1970-0257029-5 D. H. LEHMER & J. M. MASLEY, "Table of the cyclotomic class numbers ${h^\ast }(p)$ and their factors for $200 < p < 512$." (To appear.)
Additional Information
- © Copyright 1977 American Mathematical Society
- 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