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

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

Zbl 0629.12004
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