Ambiguous classes in quadratic fields. (English) Zbl 0790.11076

Mollin, R. A.

Math. Comput. 61, No. 203, 355-360 (1993).

Sei \(K\) ein reell-quadratischer Zahlkörper mit Diskriminante \(\Delta\), seien \(q_ 1,\dots,q_ n\) paarweise teilerfremde quadratfreie Teiler von \(\Delta\) und \(Q_ 1,\dots,Q_ n\) deren Primteiler in \(K\). Sei \[ F_ i(x)= q_ i x^ 2+ (\alpha_ i- 1)q_ i x+(4q_ i)^{- 1}(\alpha_ i q^ 2_ i- q^ 2_ i-\Delta) \] mit \(\alpha_ i=1\), falls \(4q_ i|\Delta\) und \(\alpha_ i=2\) sonst. Der Autor beweist ein Kriterium folgenden Typs: Stellen die quadratischen Polynome \(F_ i(z)\) hinreichend viele Primzahlen dar, so besteht die Klassengruppe von \(K\) nur aus den Klassen von \(1,Q_ 1,\dots,Q_ n\).

Reviewer: F.Halter-Koch (Graz)

MSC:

11R11	Quadratic extensions
11R29	Class numbers, class groups, discriminants
11Y40	Algebraic number theory computations

Keywords:

real quadratic fields; class group; quadratic polynomials

References:

[1]	F. Halter-Koch, Prime-producing quadratic polynomials and class numbers of quadratic orders, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 73 – 82. · Zbl 0728.11049
[2]	S. Louboutin, R. A. Mollin, and H. C. Williams, Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials and quadratic residue covers, Canad. J. Math. 44 (1992), no. 4, 824 – 842. · Zbl 0771.11039
[3]	R. A. Mollin and H. C. Williams, Prime producing quadratic polynomials and real quadratic fields of class number one, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 654 – 663.
[4]	R. A. Mollin and H. C. Williams, On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J. 112 (1988), 143 – 151. · Zbl 0629.12004
[5]	R. A. Mollin and H. C. Williams, Solution of the class number one problem for real quadratic fields of extended Richaud-Degert type (with one possible exception), Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990, pp. 417 – 425.