Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the $ 3$-rank of quadratic fields and the Euler product

Authors: Carol Neild and Daniel Shanks
Journal: Math. Comp. 28 (1974), 279-291
MSC: Primary 12A25; Secondary 12A65
MathSciNet review: 0352042
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper covers many (closely related) topics: the distribution of the 3-Sylow subgroups of imaginary quadratic fields; the possibility of finding 3-ranks greater than 4; some questions concerning $ {a^3} = {b^2} + {c^2}D$; and the convergence of Euler products and its relation to the extended Riemann hypothesis. Two programs that were used in this investigation are described.

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

  • [1] Daniel Shanks & Peter Weinberger, "A quadratic field of prime discriminant requiring three generators for its class group, and related theory," Sierpiński Memorial Volume, Acta Arith., v. 21, 1972, pp. 71-87. MR 0309899 (46:9003)
  • [2] Daniel Shanks, "New types of quadratic fields having three invariants divisible by 3," J. Number Theory, v. 4, 1972, pp. 537-556. MR 0313220 (47:1775)
  • [3] Daniel Shanks & Richard Serafin, "Quadratic fields with four invariants divisible by 3," Math. Comp., v. 27, 1973, pp. 183-187. Corrigendum, ibid., p. 1012. MR 0330097 (48:8436a)
  • [4] Peter Roquette, "On class field towers," Algebraic Number Theory, (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 231-249. MR 36 #1418. MR 0218331 (36:1418)
  • [5] H. Davenport & H. Heilbronn, "On the density of discriminants of cubic fields," Bull. London Math. Soc., v. 1, 1969, pp. 345-348. MR 40 #7223. MR 0254010 (40:7223)
  • [6] H. Davenport & H. Heilbronn, "On the density of discriminants of cubic fields. II," Proc. Roy. Soc. London Ser. A, v. 322, 1971, pp. 405-420. MR 0491593 (58:10816)
  • [7] Daniel Shanks, "Class number, a theory of factorization, and genera," Proc. Sympos. Pure Math., vol. 20, Amer. Math. Soc., Providence, R.I., 1971, pp. 415-440. MR 0316385 (47:4932)
  • [8] A. Scholz, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," Crelle's J., v. 166, 1932, pp. 201-203.
  • [9] L. J. Mordell, Diophantine Equations, Pure and Appl. Math., vol. 30, Academic Press, New York and London, 1969, Chapter 26. MR 40 #2600. MR 0249355 (40:2600)
  • [10] G. Gras, "Extensions abéliennes non ramifiées de degré premier d'un corps quadratique," Bull. Soc. Math. France, v. 100, 1972, pp. 177-193. MR 0302604 (46:1748)
  • [11] Carol C. Neild, SPEEDY, A Code for Estimating the Euler Product of a Dirichlet L Function, CMD-8-73, 1973, Naval Ship R&D Center, Bethesda, Maryland.
  • [12] Daniel Shanks, "Five number-theoretic algorithms," Proceedings of the Manitoba Conference on Numerical Mathematics, 1972, University of Manitoba, Winnipeg, Canada, 1973. MR 0371855 (51:8072)
  • [13] Richard H. Serafin, Two Subroutines for the Solution of $ R \equiv {A^H}$ (modulo N) and $ {R^2} \equiv A$ (modulo P) and their Applications, CMD-7-73, 1973, Naval Ship R&D Center, Bethesda, Maryland.
  • [14] Carol C. Neild, CUROID, A Code for Computing the Cube Roots of the Identity of the Class Group of an Imaginary Quadratic Field, CMD-9-73, 1973, Naval Ship R&D Center, Bethesda, Maryland.
  • [15] Daniel Shanks, "The infrastructure of a real quadratic field and its applications," Proceedings of the 1972 Number Theory Conference, University of Colorado, Boulder, Colorado, 1973, pp. 217-224. MR 0389842 (52:10672)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 12A25, 12A65

Retrieve articles in all journals with MSC: 12A25, 12A65

Additional Information

Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society