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

Abstract Let p ≡ 3 ( mod 4 ) {p\equiv 3\pmod{4}} and l ≡ 5 ( mod 8 ) {l\equiv 5\pmod{8}} be different primes such that p l = 1 {\frac{p}{l}=1} and 2 p = p l 4 {\frac{2}{p}=\frac{p}{l}_{4}} . Put k = ℚ ⁢ ( l ) {k=\mathbb{Q}(\sqrt{l})} , and denote by ϵ its fundamental unit. Set K = k ⁢ ( - 2 ⁢ p ⁢ ϵ ⁢ l ) {K=k(\sqrt{-2p\epsilon\sqrt{l}})} , and let K 2 ( 1 ) {K_{2}^{(1)}} be its Hilbert 2-class field, and let K 2 ( 2 ) {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 ) {(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 = Gal ⁡ ( K 2 ( 2 ) / K ) {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 ) {K_{2}^{(1)}} . Additionally, these extensions are constructed, and their abelian-type invariants are given.