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

HTML articles powered by AMS MathViewer

- by Carol Neild and Daniel Shanks PDF
- Math. Comp.
**28**(1974), 279-291 Request permission

## 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

- 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**309899**, DOI 10.4064/aa-21-1-71-87 - Daniel Shanks,
*New types of quadratic fields having three invariants divisible by $3$*, J. Number Theory**4**(1972), 537–556. MR**313220**, DOI 10.1016/0022-314X(72)90027-3 - Daniel Shanks and Richard Serafin,
*Quadratic fields with four invariants divisible by $3$*, Math. Comp.**27**(1973), 183–187. MR**330097**, DOI 10.1090/S0025-5718-1973-0330097-0 - Peter Roquette,
*On class field towers*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 231–249. MR**0218331** - H. Davenport and H. Heilbronn,
*On the density of discriminants of cubic fields*, Bull. London Math. Soc.**1**(1969), 345–348. MR**254010**, DOI 10.1112/blms/1.3.345 - 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**491593**, DOI 10.1098/rspa.1971.0075 - 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**
A. Scholz, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," - L. J. Mordell,
*Diophantine equations*, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR**0249355** - Georges Gras,
*Extensions abéliennes non ramifiées de degré premier d’un corps quadratique*, Bull. Soc. Math. France**100**(1972), 177–193 (French). MR**302604**
Carol C. Neild, - Daniel Shanks,
*Five number-theoretic algorithms*, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Congressus Numerantium, No. VII, Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. MR**0371855**
Richard H. Serafin, - 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**

*Crelle’s J.*, v. 166, 1932, pp. 201-203.

*SPEEDY, A Code for Estimating the Euler Product of a Dirichlet L Function*, CMD-8-73, 1973, Naval Ship R&D Center, Bethesda, Maryland.

*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. 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.

## Additional Information

- © Copyright 1974 American Mathematical Society
- Journal: Math. Comp.
**28**(1974), 279-291 - MSC: Primary 12A25; Secondary 12A65
- DOI: https://doi.org/10.1090/S0025-5718-1974-0352042-5
- MathSciNet review: 0352042