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.

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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document