Class groups of quadratic fields

Author:
Duncan A. Buell

Journal:
Math. Comp. **30** (1976), 610-623

MSC:
Primary 12A50; Secondary 12A25

MathSciNet review:
0404205

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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 3-Sylow subgroup are , class group a cyclic group of order *n*), and , class group . The author has also found polynomials representing discriminants of 3-rank , and has found 3-rank 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 3-rank 3.

The smallest examples of noncyclic 13-, 17-, and 19-Sylow subgroups have been found, and of groups noncyclic in two odd *p*-Sylow 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****[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****[3]**Daniel Shanks,*New types of quadratic fields having three invariants divisible by 3*, J. Number Theory**4**(1972), 537–556. MR**0313220****[4]**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**[5]**Carol Neild and Daniel Shanks,*On the 3-rank of quadratic fields and the Euler product*, Math. Comp.**28**(1974), 279–291. MR**0352042**, 10.1090/S0025-5718-1974-0352042-5**[6]**F. DIAZ Y DIAZ, "Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supérieur à 1,"*Séminaire Delange-Pisot-Poitou*, 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**, 10.1090/S0025-5718-1976-0399039-9**[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****[8]**ARNOLD SCHOLZ, "Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,"*J. Reine Angew. Math.*, v. 166, 1932, pp. 201-203.**[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****[10]**R. A. Lippmann,*Note on irregular discriminatnts*, J. London Math. Soc.**38**(1963), 385–386. MR**0157946****[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. 491-493.**[14]**Hideo Wada,*A table of ideal class groups of imaginary quadratic fields*, Proc. Japan Acad.**46**(1970), 401–403. MR**0366866****[15]**S. CHOWLA, "An extension of Heilbronn's class number theorem,"*Quart. J. Math. Oxford Ser.*, v. 5, 1934, pp. 304-307.**[16]**W. Narkiewicz,*Class number and factorization in quadratic number fields*, Colloq. Math.**17**(1967), 167–190. MR**0220698****[17]**Daniel Shanks,*On Gauss’s class number problems*, Math. Comp.**23**(1969), 151–163. MR**0262204**, 10.1090/S0025-5718-1969-0262204-1

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1976-0404205-X

Article copyright:
© Copyright 1976
American Mathematical Society