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Capitulation of 2-ideal classes of certain cyclic biquadratic fields. (Capitulation des 2-classes d’idĂ©aux de certains corps biquadratiques cycliques.) (French. English summary) Zbl 1169.11049
Summary: Let \(K=k(\sqrt{-pq{\varepsilon}\sqrt 2})\) with \(k=\mathbb Q(\sqrt 2)\), \(\varepsilon=1+\sqrt 2\) the fundamental unit of \(k\), \(p\) and \(q\) two different prime numbers such that \(p\equiv q\equiv \pm 1\bmod 4\) and \(\big(\frac{2}{p}\big)=\big(\frac{2}{q}\big)=-1\), \(K_2^{(1)}\) be the Hilbert \(2\)-class field of \(K\), \(K_2^{(2)}\) be the Hilbert \(2\)-class field of \(K_2^{(1)}\) and \(G=\text{Gal}(K_2^{(2)}/K)\) be the Galois group of \(K_2^{(2)}/K\). According to E. Brown and C. J. Parry [Pac. J. Math. 78, 11–26 (1978; Zbl 0405.12009)], \(C_{2,K}\), the \(2\)-part of the ideal class group of \(K\), is isomorphic to \(\mathbb Z/{2\mathbb Z}\times \mathbb Z/{2\mathbb Z}\), consequently \(K_2^{(1)}/K\) contains three extensions \(F_i/K\) \((i=1,2,3)\) and the tower of the Hilbert \(2\)-class field of \(K\) terminates at either \(K_2^{(1)}\) or \(K_2^{(2)}\). In this paper, we study the problem of capitulation of the classes of \(C_{2,K}\) in \(F_i\) \((i=1,2,3)\) and we determine the structure of \(G\).

11R37 Class field theory
11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions
Zbl 0405.12009
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