Abdelmalek Azizi, Idriss Jerrari, Abdelkader Zekhnini, Mohammed Talbi
October 12, 2018

### Abstract

Let p≡3(mod4){p\equiv 3\pmod{4}} and l≡5(mod8){l\equiv 5\pmod{8}} be different primes such that pl=1{\frac{p}{l}=1} and 2p=pl4{\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(-2pϵl){K=k(\sqrt{-2p\epsilon\sqrt{l}})}, and let K2(1){K_{2}^{(1)}} be its Hilbert 2-class field, and let K2(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(K2(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 K2(1){K_{2}^{(1)}}. Additionally, these extensions are constructed, and their abelian-type invariants are given.