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 Free Access

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 and Peter Weinberger, A quadratic field of prime discriminant requiring three generators for its class group, and related theory, Acta Arith. 21 (1972), 71–87. MR 0309899
  • [2] Daniel Shanks, New types of quadratic fields having three invariants divisible by 3, J. Number Theory 4 (1972), 537–556. MR 0313220
  • [3] Daniel Shanks and Richard Serafin, Quadratic fields with four invariants divisible by 3, Math. Comp. 27 (1973), 183–187. MR 0330097, 10.1090/S0025-5718-1973-0330097-0
  • [4] Peter Roquette, On class field towers, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 231–249. MR 0218331
  • [5] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields, Bull. London Math. Soc. 1 (1969), 345–348. MR 0254010
  • [6] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 0491593
  • [7] Daniel Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR 0316385
  • [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 Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
  • [10] Georges Gras, Extensions abéliennes non ramifiées de degré premier d’un corps quadratique, Bull. Soc. Math. France 100 (1972), 177–193 (French). MR 0302604
  • [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 Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. Congressus Numerantium, No. VII. MR 0371855
  • [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 Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 217–224. MR 0389842

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