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.

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.

On Hilbert class field tower for some quartic number fields

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohamed Talbi ◽  
Mohammed Talbi
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Class Field ◽  
Hilbert Class Field ◽  
Class Field Tower

We determine the Hilbert 2-class field tower for some quartic number fields k whose 2-class group Ck,2 is isomorphic to ℤ/2ℤ×ℤ/2ℤ.

DISTRIBUTION OF GALOIS GROUPS OF MAXIMAL UNRAMIFIED 2-EXTENSIONS OVER IMAGINARY QUADRATIC FIELDS

2018 ◽  
Vol 237 ◽  
pp. 166-187
Author(s):  
SOSUKE SASAKI
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Class Field ◽  
Class Field Towers ◽  
Generalized Quaternion ◽  
Class Field Tower

Let $k$ be an imaginary quadratic field with $\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert $2$-class field tower is at least $2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J. 40(1) (1998), 63–69) calculated the density of $k$ where the length of the tower is $1$; that is, the maximal unramified $2$-extension is a $V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

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 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.

On the 2-class field tower of subfields of some cyclotomic ℤ2-extensions

2019 ◽  
Vol 69 (1) ◽  
pp. 81-86
Author(s):  
Ali Mouhib
Keyword(s):  
Positive Integer ◽  
Galois Group ◽  
Large Degree ◽  
Number Fields ◽  
Class Field ◽  
Class Field Tower

Abstract We study the structure of the Galois group of the maximal unramified 2-extension of some family of number fields of large degree. Especially, we show that for each positive integer n, there exist infinitely many number fields with large degree, for which the defined Galois group is quaternion of order 2n.

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].

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.

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