Groupe des classes d’idéaux triviaux. (Class group of trivial ideals). (French) Zbl 0634.12008

This paper is a generalization of our previous one: “Continued fractions and real quadratic fields” (to appear in J. Number Theor.). In it we give two necessary and sufficient conditions for the ideal class group \({\mathcal H}\) of a real quadratic field with character x to be reduced to its subgroup \({\mathcal A}\) of ambiguous classes. The first one is a generalization of the Frobenius-Rabinovich theorem. In order to apply these criteria we give first a continued fractional proof of the well known result about the order of \({\mathcal A}\) and then apply them to the family of real quadratic fields \({\mathbb{Q}}(\sqrt{d})\), \(d=m^ 2-1\) square free. We get the equivalence of the three following assertions:
(a) m-1 or \(m+1\) is prime and \({\mathcal H}={\mathcal A}.\)
(b) \(3\leq p\leq m-1\) and p prime imply \((d/p)=x(p)\neq +1.\)
(c) The divisors of \(d-k^ 2\), \(1\leq k\leq m-1\), which are odd and less than or equal to m-1 divide d.
This leads naturally to the study of the family \({\mathcal F}=\{{\mathbb{Q}}(\sqrt{d})\); d square free and \(x(p)\neq +1\), \(2\leq p\leq ()\sqrt{D}\}\) for which \({\mathcal H}={\mathcal A}\). Under Riemann’s hypothesis \(\zeta _{{\mathbb{Q}}(\sqrt{d})/{\mathbb{Q}}}()\leq 0\) we are able to give explicitly the 60 fields of this family.
Reviewer: L.Louboutin


11R23 Iwasawa theory
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