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On some metabelian 2-groups and applications I

Abdelmalek AziziAbdelkader ZekhniniMohammed Taous — 2016

Colloquium Mathematicae

Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition G a l ( k ( 2 ) / k ) G , where k ( 2 ) is the second Hilbert 2-class field of k.

Sur un problème de capitulation du corps ( p 1 p 2 , i ) dont le 2 -groupe de classes est élémentaire

Abdelmalek AziziAbdelkader ZekhniniMohammed Taous — 2014

Czechoslovak Mathematical Journal

Soient p 1 p 2 1 ( mod 8 ) des nombres premiers tels que, ( p 1 p 2 ) = - 1 et ( 2 a + b ) = - 1 , où p 1 p 2 = a 2 + b 2 . Soient i = - 1 , d = p 1 p 2 , 𝕜 = ( d , i ) , 𝕜 2 ( 1 ) le 2-corps de classes de Hilbert de 𝕜 et 𝕜 ( * ) = ( p 1 , p 2 , i ) le corps de genres de 𝕜 . La 2-partie C 𝕜 , 2 du groupe de classes de 𝕜 est de type ( 2 , 2 , 2 ) , par suite 𝕜 2 ( 1 ) contient sept extensions quadratiques non ramifiées 𝕂 j / 𝕜 et sept extensions biquadratiques non ramifiées 𝕃 j / 𝕜 . Dans ce papier on s’intéresse à déterminer ces quatorze extensions, le groupe C 𝕜 , 2 et à étudier la capitulation des 2-classes d’idéaux de 𝕜 dans ces extensions.

On the strongly ambiguous classes of some biquadratic number fields

Abdelmalek AziziAbdelkader ZekhniniMohammed Taous — 2016

Mathematica Bohemica

We study the capitulation of 2 -ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields 𝕜 = ( 2 p q , i ) , where i = - 1 and p - q 1 ( mod 4 ) are different primes. For each of the three quadratic extensions 𝕂 / 𝕜 inside the absolute genus field 𝕜 ( * ) of 𝕜 , we determine a fundamental system of units and then compute the capitulation kernel of 𝕂 / 𝕜 . The generators of the groups Am s ( 𝕜 / F ) and Am ( 𝕜 / F ) are also determined from which we deduce that 𝕜 ( * ) is smaller than the relative genus field ( 𝕜 / ( i ) ) * . Then we prove that each...

On the 2 -class group of some number fields with large degree

Let d be an odd square-free integer, m 3 any integer and L m , d : = ( ζ 2 m , d ) . In this paper, we shall determine all the fields L m , d having an odd class number. Furthermore, using the cyclotomic 2 -extensions of some number fields, we compute the rank of the 2 -class group of L m , d whenever the prime divisors of d are congruent to 3 or 5 ( mod 8 ) .

On the Hilbert 2 -class field tower of some imaginary biquadratic number fields

Mohamed Mahmoud Chems-EddinAbdelmalek AziziAbdelkader ZekhniniIdriss Jerrari — 2021

Czechoslovak Mathematical Journal

Let 𝕜 = 2 , d be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and 𝕜 2 ( 2 ) its second Hilbert 2 -class field. Denote by G = Gal ( 𝕜 2 ( 2 ) / 𝕜 ) the Galois group of 𝕜 2 ( 2 ) / 𝕜 . The purpose of this note is to investigate the Hilbert 2 -class field tower of 𝕜 and then deduce the structure of G .

Note on the Hilbert 2-class field tower

Let k be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields 𝕜 = ( d , - 1 ) , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

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