The number of real quadratic fields having units of negative norm. (English) Zbl 0792.11041

Stevenhagen, Peter

Exp. Math. 2, No. 2, 121-136 (1993).

The author considers the density of discriminants of quadratic fields where \(N(\alpha)=-1\) is solvable (negative Pell equation) over those fields where \(N(\alpha/\beta) =-1\) is solvable, \(\alpha,\beta\in {\mathcal O}\) (discriminants not divisible by “bad” primes \(\equiv -1\bmod 4\)). The density is conjectured to be \(.580577\cdots =1- \prod (1-1/2^ j)\), (odd \(j>0\)). Experimental evidence for the density is higher because, practically, the number of factors in the discriminant is small. The answer is based on the density of the maximum-rank Redei matrices of relative residues of the discriminant divisors, assuming uniform distribution of residuacity.

Reviewer: H.Cohn (New York)

MSC:

11R11	Quadratic extensions
11R29	Class numbers, class groups, discriminants
11D09	Quadratic and bilinear Diophantine equations

Keywords:

4-class group; density of discriminants of quadratic fields; Pell equation; Redei matrices

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