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Capitulation in certain unramified extensions of cyclic quartic fields. (Capitulation dans certaines extensions non ramifi√©es 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
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