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

• Main Authors: Azizi Abdelmalek, Jerrari Idriss, Zekhnini Abdelkader, Talbi Mohammed
• Format: Article
• Language: English
• Published: De Gruyter 2019-03-01
• Series: Journal of Mathematical Cryptology
• Subjects: capitulation; hilbert class field; 11r11; 11r16; 11r29; 11r32; 11r37
• Online Access: https://doi.org/10.1515/jmc-2017-0037
• ISSN: 1862-2976; 1862-2984

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⁢(-2⁢p⁢ϵ⁢l){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.