Principalization of 2-class groups of type (2, 2, 2) of biquadratic fields ${\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$

2015 ◽  
Vol 11 (04) ◽  
pp. 1177-1215 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous ◽  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Nilpotency Class ◽  
Class Groups ◽  
Class Field ◽  
Abelian Extensions ◽  
Biquadratic Fields

Let p1 ≡ p2≡ -q ≡ 1 (mod 4) be different primes such that [Formula: see text]. Put d = p1p2q and [Formula: see text], then the bicyclic biquadratic field [Formula: see text] has an elementary abelian 2-class group, Cl2(𝕜), of rank 3. In this paper, we study the principalization of the 2-classes of 𝕜 in its 14 unramified abelian extensions 𝕂j and 𝕃j within [Formula: see text], that is the Hilbert 2-class field of 𝕜. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group [Formula: see text] of the second Hilbert 2-class field [Formula: see text] of 𝕂. Additionally, the abelian type invariants of the groups Cl2(𝕂j) and Cl2(𝕃j) and the length of the 2-class tower of 𝕜 are given.

Coclass of ${\rm Gal}({\mathbb K}_2^{(2)}/{\mathbb K})$ for some fields ${\mathbb K} = {\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$ with 2-class groups of types (2, 2, 2)

2015 ◽  
Vol 15 (02) ◽  
pp. 1650027 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Nilpotency Class ◽  
Class Groups ◽  
Class Field ◽  
Class Field Tower

Let p1 ≡ p2 ≡ -q ≡ 1 ( mod 4) be primes such that [Formula: see text] and [Formula: see text]. Put [Formula: see text] and d = p1p2q, then the bicyclic biquadratic field [Formula: see text] has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group [Formula: see text] of the second Hilbert 2-class field [Formula: see text] of 𝕂, we study the 2-class field tower of 𝕂, and we study the capitulation problem of the 2-classes of 𝕂 in its fourteen abelian unramified extensions of relative degrees two and four.

The rational characterization of certain sets of relatively Abelian extensions

1959 ◽  
Vol 251 (998) ◽  
pp. 385-425 ◽  
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Class Group ◽  
Class Field Theory ◽  
Abelian Extension ◽  
Galois Groups ◽  
Class Field ◽  
Abelian Extensions ◽  
Rational Field

Let H be a class group— in the sense of class-field theory— in the rational field P, whose order is some power of a prime l . With H there is associated an Abelian extension K of P. The purpose of this paper is to determine in rational terms and for all fields K given in the described manner, the set T(K/P) of cyclic extensions A of K of relative degree l , which are absolutely normal. In particular we shall find the ramification laws for these fields A, and the possible extension types of a group of order l by the Galois group of K, which are realized in Galois groups of fields in T(K/P). It is fundamental to the programme outlined, that we aim at obtaining purely rational criteria of determination.

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

2019 ◽  
Vol 13 (1) ◽  
pp. 27-46
Author(s):  
Abdelmalek Azizi ◽  
Idriss Jerrari ◽  
Abdelkader Zekhnini ◽  
Mohammed Talbi
Keyword(s):  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Fundamental Unit ◽  
Class Field ◽  
Abelian Extensions ◽  
Class Field Tower

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.

The group Gal(k3(2)|k) for k = ℚ(−3,d) of type (3,3)

2016 ◽  
Vol 12 (07) ◽  
pp. 1951-1986 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohamed Talbi ◽  
Mohammed Talbi ◽  
Aïssa Derhem ◽  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Quadratic Field ◽  
Galois Groups ◽  
Numerical Verification ◽  
Real Quadratic Field ◽  
Class Field ◽  
Type Formula ◽  
Biquadratic Fields ◽  
Real Quadratic

Let [Formula: see text] denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields [Formula: see text], having a [Formula: see text]-class group of type [Formula: see text], the possibilities for the isomorphism type of the Galois group [Formula: see text] of the second Hilbert [Formula: see text]-class field [Formula: see text] of [Formula: see text] are determined. For each coclass graph [Formula: see text], [Formula: see text], in the sense of Eick, Leedham-Green, Newman and O’Brien, the roots [Formula: see text] of even branches of exactly one coclass tree and, in the case of even coclass [Formula: see text], additionally their siblings of depth [Formula: see text] and defect [Formula: see text], turn out to be admissible. The principalization type [Formula: see text] of [Formula: see text]-classes of [Formula: see text] in its four unramified cyclic cubic extensions [Formula: see text] is given by [Formula: see text] for [Formula: see text], and by [Formula: see text] for [Formula: see text]. The theory is underpinned by an extensive numerical verification for all [Formula: see text] fields [Formula: see text] with values of [Formula: see text] in the range [Formula: see text], which supports the assumption that all admissible vertices [Formula: see text] will actually be realized as Galois groups [Formula: see text] for certain fields [Formula: see text], asymptotically.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

scholarly journals THE SECOND p-CLASS GROUP OF A NUMBER FIELD

2012 ◽  
Vol 08 (02) ◽  
pp. 471-505 ◽  
Author(s):  
DANIEL C. MAYER
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Number Field ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Green Is ◽  
Class Field ◽  
Type 3

For a prime p ≥ 2 and a number field K with p-class group of type (p, p) it is shown that the class, coclass, and further invariants of the metabelian Galois group [Formula: see text] of the second Hilbert p-class field [Formula: see text] of K are determined by the p-class numbers of the unramified cyclic extensions Ni|K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field [Formula: see text] and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields Li|ℚ of absolute degree p of the dihedral fields Ni. As an application, the structure of the automorphism group [Formula: see text] of the second Hilbert 3-class field [Formula: see text] is analyzed for all quadratic fields K with discriminant -106 < D < 107 and 3-class group of type (3, 3) by computing their principalization types. The distribution of these metabelian 3-groups G on the coclass graphs [Formula: see text], 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated.

scholarly journals 2-Class groups of cyclotomic towers of imaginary biquadratic fields and applications

2021 ◽  
pp. 1-13
Author(s):  
Mohamed Mahmoud Chems-Eddin ◽  
Katharina Müller
Keyword(s):  
Number Field ◽  
Class Group ◽  
Class Groups ◽  
Biquadratic Fields

Let [Formula: see text] be an odd positive square-free integer. In this paper, we shall investigate the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of the imaginary biquadratic number field [Formula: see text] if [Formula: see text] is of specifiic form. Furthermore, we deduce the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of [Formula: see text].

The structure of the second 2-class group of some special Dirichlet fields

2018 ◽  
Vol 13 (03) ◽  
pp. 2050053
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Class Group ◽  
Class Field

Let [Formula: see text] be a number field and [Formula: see text] (respectively, [Formula: see text]) its first (respectively, second) Hilbert [Formula: see text]-class field. Let [Formula: see text] be the Galois group of [Formula: see text]. The purpose of this note is to determine the structure of [Formula: see text] for some special Dirichlet fields [Formula: see text].

LE 2-RANG DU GROUPE DE CLASSES DE CERTAINS CORPS BIQUADRATIQUES ET APPLICATIONS

2004 ◽  
Vol 15 (02) ◽  
pp. 169-182
Author(s):  
ABDELMALEK AZIZI ◽  
ALI MOUHIB
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Biquadratic Fields ◽  
Class Field Tower ◽  
Real Quadratic

Let K be a real biquadratic field and let k be a quadratic field with odd class number contained in K. The aim of this article is to determine the rank of the 2-class group of K and we give applications to the structure of the 2-class group of some biquadratic fields and to the 2-class field tower of some real quadratic fields. Résumé: Soient K un corps biquadratique réel et k un sous-corps quadratique de K dont le nombre de classes est impair. Dans ce papier on détermine le rang du 2-groupe de classes de K et on donne des applications à la structure du 2-groupe de classes de certains corps biquadratiques et aussi à la tour des 2-corps de classes de Hilbert de certains corps quadratiques réels.

The 2-Rank of the Class Group of Imaginary Bicyclic Biquadratic Fields

1997 ◽  
Vol 49 (2) ◽  
pp. 283-300 ◽  
Author(s):  
Thomas M. McCall ◽  
Charles J. Parry ◽  
Ramona R. Ranalli
Keyword(s):  
Sylow Subgroup ◽  
Class Group ◽  
Class Numbers ◽  
Ideal Class ◽  
Small Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Biquadratic Fields ◽  
The Ideal

AbstractA formula is obtained for the rank of the 2-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the 2-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a Z2-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both 2- class and class group Z2×Z2 are determined. The final results assume the completeness of D. A. Buell’s list of imaginary fields with small class numbers.

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