On Gauss's Class Number Problems
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چکیده
Let h be the class number of binary quadratic forms (in Gauss's formulation). All negative determinants having some h = On ± 1 can be determined constructively: for h = 5 there are four such determinants; for h = 7, six; for A = 11, four; and for h = 13, six. The distinction between class numbers for determinants and for discriminants is discussed and some data are given. The question of one class/genus for negative determinants is imbedded in the larger question of the existence of a determinant having a specific Abelian group as its composition group. All Abelian groups of order <25 so exist, but the noncyclic groups of order 25, 49, and 121 do not occur. Positive determinants are treated by the same composition method. Although most positive primes of the form n2 — 8 have h = 1, an interesting subset does not. A positive determinant of an odd exponent of irregularity also appears in the investigation. Gauss indicated that he could not find one. |
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تاریخ انتشار 2010