## Ambiguous logarithmic classes of quadratic fields. (Classes logarithmiques ambiges des corps quadratiques.)(French)Zbl 0869.11081

The main subject of this article is a comparison of two events $$A_K$$ and $$B_K$$ which can occur for a quadratic extension $$K$$ of the rationals. We explain: $$A_K$$ means that the 2-component of the wild kernel (noyau hilbertien) of $$K$$ is trivial; $$B_K$$ means that the 2-component of the group of logarithmic classes (invented by J.-F. Jaulent) is trivial. Of course the two statements $$A_K$$ and $$B_K$$ make sense for any number field $$K$$; if $$\zeta_4\in K$$, they are known to be equivalent. Note that quadratic fields rarely contain $$\zeta_4$$. The wild kernel can be computed by descent and Iwasawa theory; this involves twisting by 1 before descending. If one omits the twist, one obtains Jaulent’s group instead.
The results proved for quadratic fields $$K$$ in this paper are too involved to be repeated here (see Theorem 6.1.2). One result which is easy to state is: $$A_K$$ implies $$B_K$$ (Scholium 6.1.3), and the author also shows that the converse is far from true. The main technical expedient is a formula for ambiguous logarithmic classes, similar in shape to the corresponding formula for ambiguous classes (in the usual sense), and again due to Jaulent.

### MSC:

 11R11 Quadratic extensions 11R29 Class numbers, class groups, discriminants 11R70 $$K$$-theory of global fields
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