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Capitulation of 2-ideal classes of certain cyclic biquadratic fields. (Capitulation des 2-classes d’idéaux 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 $$G$$.

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