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Capitulation in certain unramified extensions of cyclic quartic fields. (Capitulation dans certaines extensions non ramifiées de corps quartiques cycliques.) (French) Zbl 1212.11091
Summary: Let $$K=k\big (\sqrt {-p{\varepsilon }\sqrt {l}}\big )$$ with $$k={\mathbb Q}(\sqrt {l})$$ where $$l$$ is a prime number such that $$l=2$$ or $$l\equiv 5\bmod 8$$, $$\varepsilon$$ the fundamental unit of $$k$$, $$p$$ a prime number such that $$p\equiv 1\bmod 4$$ and $${(p/l)}_4=-1$$, $$K_2^{(1)}$$ the Hilbert $$2$$-class field of $$K$$, $$K_2^{(2)}$$ the Hilbert $$2$$-class field of $$K_2^{(1)}$$ and $$G=\text{Gal} (K_2^{(2)}/K)$$ the Galois group of $$K_2^{(2)}/K$$. According to E. Brown and C. J. Parry [J. Reine Angew. Math. 295, 61–71 (1977; Zbl 0355.12007) and Pac. J. Math. 78, 11–26 (1978; Zbl 0405.12009)], $$C_{2,K}$$, the Sylow $$2$$-subgroup 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 work we are interested in the problem of capitulation of the classes of $$C_{2,K}$$ in $$F_i$$ $$(i=1,2,3)$$ and determine the structure of $$G$$.

##### MSC:
 11R27 Units and factorization 11R37 Class field theory 11R16 Cubic and quartic extensions
##### Keywords:
biquadratic field; capitulation; Hilbert class field
##### Citations:
Zbl 0355.12007; Zbl 0405.12009
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