Class groups of quadratic fields
Author:
Duncan A. Buell
Journal:
Math. Comp. 30 (1976), 610623
MSC:
Primary 12A50; Secondary 12A25
MathSciNet review:
0404205
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The author has computed the class groups of all complex quadratic number fields of discriminant for . In so doing, it was found that the first occurrences of rank three in the 3Sylow subgroup are , class group a cyclic group of order n), and , class group . The author has also found polynomials representing discriminants of 3rank , and has found 3rank 3 for , all prime, , and . The first five of these were discovered by Diaz y Diaz, using a different method. The author believes, however, that his computation independently establishes the fact that 3321607 and 3640387 are the smallest D with 3rank 3. The smallest examples of noncyclic 13, 17, and 19Sylow subgroups have been found, and of groups noncyclic in two odd pSylow subgroups. , class group , had been found by A. O. L. Atkin; the next such D is , class group . Finally, has class group , the smallest D noncyclic for 3 and 7 together.
 [1]
Maurice
Craig, A type of class group for imaginary quadratic fields,
Acta Arith. 22 (1973), 449–459. (errata insert). MR 0318098
(47 #6647)
 [2]
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
(46 #9003)
 [3]
Daniel
Shanks, New types of quadratic fields having three invariants
divisible by 3, J. Number Theory 4 (1972),
537–556. MR 0313220
(47 #1775)
 [4]
Daniel
Shanks and Richard
Serafin, Quadratic fields with four invariants
divisible by 3, Math. Comp. 27 (1973), 183–187. MR 0330097
(48 #8436a), http://dx.doi.org/10.1090/S00255718197303300970
 [5]
Carol
Neild and Daniel
Shanks, On the 3rank of quadratic fields and
the Euler product, Math. Comp. 28 (1974), 279–291. MR 0352042
(50 #4530), http://dx.doi.org/10.1090/S00255718197403520425
 [6]
F. DIAZ Y DIAZ, "Sur les corps quadratiques imaginaires dont le 3rang du groupe des classes est supérieur à 1," Séminaire DelangePisotPoitou, 1973/74, no. G15.
 [6a]
Daniel
Shanks, Class groups of the quadratic fields
found by F. Diaz y Diaz, Math. Comp.
30 (1976), no. 133, 173–178. MR 0399039
(53 #2890), http://dx.doi.org/10.1090/S00255718197603990399
 [7]
Erich
Hecke, Vorlesungen über die Theorie der algebraischen
Zahlen, Chelsea Publishing Co., Bronx, N.Y., 1970 (German). Second
edition of the 1923 original, with an index. MR 0352036
(50 #4524)
 [8]
ARNOLD SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," J. Reine Angew. Math., v. 166, 1932, pp. 201203.
 [9]
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
(47 #4932)
 [10]
R.
A. Lippmann, Note on irregular discriminatnts, J. London Math.
Soc. 38 (1963), 385–386. MR 0157946
(28 #1174)
 [11]
E. T. ORDMAN, "Tables of class numbers for negative prime discriminants," UMT 29, Math. Comp., v. 23, 1969, p. 458.
 [12]
M. NEWMAN, "Table of the class number for p prime, ," UMT 50, Math. Comp., v. 23, 1969, p. 683.
 [13]
RICHARD B. LAKEIN & SIGEKATU KURODA, "Tables of class numbers for fields ," UMT 39, Math. Comp., v. 24, 1970, pp. 491493.
 [14]
Hideo
Wada, A table of ideal class groups of imaginary quadratic
fields, Proc. Japan Acad. 46 (1970), 401–403.
MR
0366866 (51 #3112)
 [15]
S. CHOWLA, "An extension of Heilbronn's class number theorem," Quart. J. Math. Oxford Ser., v. 5, 1934, pp. 304307.
 [16]
W.
Narkiewicz, Class number and factorization in quadratic number
fields, Colloq. Math. 17 (1967), 167–190. MR 0220698
(36 #3750)
 [17]
Daniel
Shanks, On Gauss’s class number
problems, Math. Comp. 23 (1969), 151–163. MR 0262204
(41 #6814), http://dx.doi.org/10.1090/S00255718196902622041
 [1]
 MAURICE CRAIG, "A type of class group for imaginary quadratic fields," Acta Arith., v. 22, 1973, pp. 449459. MR 47 #6647. MR 0318098 (47:6647)
 [2]
 DANIEL SHANKS & PETER WEINBERGER, "A quadratic field of prime discriminant requiring three generators for its class group, and related theory," Acta Arith., v. 21, 1972, pp. 7187. MR 46 #9003. MR 0309899 (46:9003)
 [3]
 DANIEL SHANKS, "New types of quadratic fields having three invariants divisible by three," J. Number Theory, v. 4, 1972, pp. 537556. MR 47 #1775. MR 0313220 (47:1775)
 [4]
 DANIEL SHANKS & RICHARD SERAFIN, "Quadratic fields with four invariants divisible by three," Math. Comp., v. 27, 1973, pp. 183187; Corrigendum, ibid., p. 1012. MR 48 #8436a, b. MR 0330097 (48:8436a)
 [5]
 CAROL NEILD & DANIEL SHANKS, "On the 3rank of quadratic fields and the Eure product," Math. Comp., v. 28, 1974, pp. 279291. MR 0352042 (50:4530)
 [6]
 F. DIAZ Y DIAZ, "Sur les corps quadratiques imaginaires dont le 3rang du groupe des classes est supérieur à 1," Séminaire DelangePisotPoitou, 1973/74, no. G15.
 [6a]
 DANIEL SHANKS, "Class groups of the quadratic fields found by Diaz y Diaz," Math. Comp., v. 30, 1976, pp. 173178. MR 0399039 (53:2890)
 [7]
 ERICH HECKE, Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea, New York, 1970. MR 50 #4524. MR 0352036 (50:4524)
 [8]
 ARNOLD SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander," J. Reine Angew. Math., v. 166, 1932, pp. 201203.
 [9]
 DANIEL SHANKS, "Class number, a theory of factorization, and genera," Proc. Sympos. Pure Math., v. 20, Amer. Math. Soc., Providence, R. I., 1971, pp. 415440. MR 47 #4932. MR 0316385 (47:4932)
 [10]
 R. A. LIPPMAN, "Note on irregular discriminants," J. London Math. Soc., v. 38, 1963, pp. 385386. MR 28 #1174. MR 0157946 (28:1174)
 [11]
 E. T. ORDMAN, "Tables of class numbers for negative prime discriminants," UMT 29, Math. Comp., v. 23, 1969, p. 458.
 [12]
 M. NEWMAN, "Table of the class number for p prime, ," UMT 50, Math. Comp., v. 23, 1969, p. 683.
 [13]
 RICHARD B. LAKEIN & SIGEKATU KURODA, "Tables of class numbers for fields ," UMT 39, Math. Comp., v. 24, 1970, pp. 491493.
 [14]
 H. WADA, "A table of ideal class groups of imaginary quadratic fields," Proc. Japan Acad., v. 46, 1970, pp. 401403. MR 0366866 (51:3112)
 [15]
 S. CHOWLA, "An extension of Heilbronn's class number theorem," Quart. J. Math. Oxford Ser., v. 5, 1934, pp. 304307.
 [16]
 W. NARKIEWICZ, "Class number and factorization in quadratic number fields," Colloq. Math., v. 17, 1967, pp. 167190. MR 36 #3750. MR 0220698 (36:3750)
 [17]
 DANIEL SHANKS, "On Gauss's classnumber problems," Math Comp., v. 23, 1969, pp. 151163. MR 41 #6814. MR 0262204 (41:6814)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
12A50,
12A25
Retrieve articles in all journals
with MSC:
12A50,
12A25
Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819760404205X
PII:
S 00255718(1976)0404205X
Article copyright:
© Copyright 1976
American Mathematical Society
