Quadratic fields with -rank equal to

Authors:
F. Diaz y Diaz, Daniel Shanks and H. C. Williams

Journal:
Math. Comp. **33** (1979), 836-840

MSC:
Primary 12A25; Secondary 12A50

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521299-4

MathSciNet review:
521299

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Abstract: In [2] there is reference to 119 known imaginary quadratic fields that have 3-rank . We examine these fields and determine the exact values of *r*. Their associated real fields and the distribution of their 3-Sylow subgroups are also studied. Some of the class groups are recorded since they are of special interest. These include examples having an infinite class field tower and only one ramified prime, and others having an infinite tower because of two different components of their class groups.

**[1]**A. SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,"*Crelle's J.*, v. 166, 1932, pp. 201-203.**[2]**F. Diaz y Diaz,*On some families of imaginary quadratic fields*, Math. Comp.**32**(1978), no. 142, 637–650. MR**0485775**, https://doi.org/10.1090/S0025-5718-1978-0485775-4**[3]**Francisco Diaz y Diaz,*Sur le 3-rang des corps quadratiques*, Publications Mathématiques d’Orsay 78, vol. 11, Université de Paris-Sud, Département de Mathématique, Orsay, 1978 (French). MR**532177****[4]**ROBERT BURNS, "The best laid schemes/O' mice an' men/Gang aft a-gley."**[5]**DANIEL SHANKS, "A matrix underlying the composition of quadratic forms and its implications for cubic extensions,"*Notices Amer. Math. Soc.*, v. 25, 1978, p. A305.**[6]**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****[7]**Daniel Shanks and Richard Serafin,*Quadratic fields with four invariants divisible by 3*, Math. Comp.**27**(1973), 183–187. MR**0330097**, https://doi.org/10.1090/S0025-5718-1973-0330097-0**[8]**Carol Neild and Daniel Shanks,*On the 3-rank of quadratic fields and the Euler product*, Math. Comp.**28**(1974), 279–291. MR**0352042**, https://doi.org/10.1090/S0025-5718-1974-0352042-5**[9]**JAMES J. SOLDERITSCH,*Imaginary Quadratic Number Fields with Special Class Groups*, Thesis, Lehigh University, 1977.**[10]**Daniel Shanks,*New types of quadratic fields having three invariants divisible by 3*, J. Number Theory**4**(1972), 537–556. MR**0313220**, https://doi.org/10.1016/0022-314X(72)90027-3**[11]**Maurice Craig,*A construction for irregular discriminants*, Osaka J. Math.**14**(1977), no. 2, 365–402. MR**0450226****[12]**MAURICE CRAIG,*Irregular Discriminants*, Dissertation, University of Michigan, Ann Arbor, Mich., 1972.

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0521299-4

Article copyright:
© Copyright 1979
American Mathematical Society