## 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

### MSC:

 11R23 Iwasawa theory

### Keywords:

ambiguous ideal; class group
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