On the rank of quadratic fields and the Euler product
Authors:
Carol Neild and Daniel Shanks
Journal:
Math. Comp. 28 (1974), 279291
MSC:
Primary 12A25; Secondary 12A65
MathSciNet review:
0352042
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Abstract 
References 
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Abstract: This paper covers many (closely related) topics: the distribution of the 3Sylow subgroups of imaginary quadratic fields; the possibility of finding 3ranks greater than 4; some questions concerning ; and the convergence of Euler products and its relation to the extended Riemann hypothesis. Two programs that were used in this investigation are described.
 [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. 7187. 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. 537556. MR 0313220 (47:1775)
 [3]
Daniel Shanks & Richard Serafin, "Quadratic fields with four invariants divisible by 3," Math. Comp., v. 27, 1973, pp. 183187. 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. 231249. 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. 345348. 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. 405420. 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. 415440. MR 0316385 (47:4932)
 [8]
A. Scholz, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," Crelle's J., v. 166, 1932, pp. 201203.
 [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. 177193. MR 0302604 (46:1748)
 [11]
Carol C. Neild, SPEEDY, A Code for Estimating the Euler Product of a Dirichlet L Function, CMD873, 1973, Naval Ship R&D Center, Bethesda, Maryland.
 [12]
Daniel Shanks, "Five numbertheoretic 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 (modulo N) and (modulo P) and their Applications, CMD773, 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, CMD973, 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. 217224. MR 0389842 (52:10672)
 [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. 7187. 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. 537556. MR 0313220 (47:1775)
 [3]
 Daniel Shanks & Richard Serafin, "Quadratic fields with four invariants divisible by 3," Math. Comp., v. 27, 1973, pp. 183187. 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. 231249. 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. 345348. 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. 405420. 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. 415440. MR 0316385 (47:4932)
 [8]
 A. Scholz, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," Crelle's J., v. 166, 1932, pp. 201203.
 [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. 177193. MR 0302604 (46:1748)
 [11]
 Carol C. Neild, SPEEDY, A Code for Estimating the Euler Product of a Dirichlet L Function, CMD873, 1973, Naval Ship R&D Center, Bethesda, Maryland.
 [12]
 Daniel Shanks, "Five numbertheoretic 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 (modulo N) and (modulo P) and their Applications, CMD773, 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, CMD973, 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. 217224. MR 0389842 (52:10672)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403520425
PII:
S 00255718(1974)03520425
Article copyright:
© Copyright 1974
American Mathematical Society
