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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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