Classification and enumeration of real quadratic fields having exactly one non-inert prime less than a Minkowski bound. (English) Zbl 0803.11054

Mollin, R. A.; Williams, H. C.

Can. Math. Bull. 36, No. 1, 108-115 (1993).

For a positive square-free integer \(d\), we denote by \(\Delta_ d\) the discriminant of the real quadratic field \(\mathbb{Q} (\sqrt{d})\). In [Nagoya Math. J. 112, 143-151 (1988; Zbl 0629.12004)], the authors classified all those real quadratic fields for which there were no non-inert primes less than \(\sqrt{\Delta_ d}/2\) in connection with the class number one problem, and showed that these fields are of narrow Richaud-Degert type: i.e. of the form \(d= m^ 2+r\) where \(| r|=1\) or 4.
In this paper, they intend to determine all \(d\)’s for which there is exactly one non-inert prime less than \(\sqrt {\Delta_ d} /2\), and list 63 values, of all such \(d\) with one possible exceptional value remaining (whose existence would be a counterexample to the extended Riemann hypothesis).

Reviewer: H.Yokoi (Nagoya)

MSC:

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

Keywords:

ideal class group; discriminant; real quadratic fields; non-inert prime

Citations:

Zbl 0629.12004