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Bi-quadratic number fields with trivial 2-primary Hilbert kernels. (English) Zbl 1058.11065

The authors completely determine all those bi-quadratic extensions of \(\mathbb{Q}\) whose wild kernel (Hilbert kernel) has trivial 2-primary part. In order to achieve this they state and prove a genus formula for the 2-primary part of the wild kernel in a relative quadratic extension. This result is of considerable independent interest. A version of this formula was stated, without proof, in the paper [Ann. Inst. Fourier 50, No. 1, 35–65 (2000; Zbl 0951.11029)] by the same authors. This genus formula together with results of P. E. Conner and J. Hurrelbrink [J. Number Theory 88, No. 2, 263–282 (2001; Zbl 0985.11059)] and J. Hurrelbrink and M. Kolster [J. Reine Angew. Math. 499, 145–188 (1998; Zbl 1044.11100)] is used to obtain a complete list of complex bi-quadratic fields with trivial 2-primary wild kernel. For real bi-quadratic extensions, the authors use Brauer relations among the zeta functions of the field and its quadratic subfields together with the Birch-Tate conjecture, now known for abelian fields as a consequence of the work of Wiles on the Main Conjecture of Iwasawa Theory.

MSC:

11R70 \(K\)-theory of global fields
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
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