Capitulation of the 2-ideal classes of type (2, 2, 2) of some quartic cyclic number fields

Abstract

Abstract Let {p\equiv 3\pmod{4}} and {l\equiv 5\pmod{8}} be different primes such that {\frac{p}{l}=1} and {\frac{2}{p}=\frac{p}{l}_{4}} . Put {k=\mathbb{Q}(\sqrt{l})} , and denote by ϵ its fundamental unit. Set {K=k(\sqrt{-2p\epsilon\sqrt{l}})} , and let {K_{2}^{(1)}} be its Hilbert 2-class field, and let {K_{2}^{(2)}} be its second Hilbert 2-class field. The field K is a cyclic quartic number field, and its 2-class group is of type {(2,2,2)} . Our goal is to prove that the length of the 2-class field tower of K is 2, to determine the structure of the 2-group {G=\operatorname{Gal}(K_{2}^{(2)}/K)} , and thus to study the capitulation of the 2-ideal classes of K in all its unramified abelian extensions within {K_{2}^{(1)}} . Additionally, these extensions are constructed, and their abelian-type invariants are given.