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.

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.

scholarly journals Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields

1993 ◽  
Vol 48 (3) ◽  
pp. 379-383 ◽  
Author(s):  
Elliot Benjamin
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Class Field ◽  
Imaginary Quadratic Number ◽  
Hilbert Class Field ◽  
Class Field Tower ◽  
The Ideal

Letkbe an imaginary quadratic number field and letk1be the 2-Hilbert class field ofk. IfCk,2, the 2-Sylow subgroup of the ideal class group ofk, is elementary and |Ck,2|≥ 8, we show thatCk1,2is not cyclic. IfCk,2is isomorphic toZ/2Z×Z/4ZandCk1,2is elementary we show thatkhas finite 2-class field tower of length at most 2.

scholarly journals On 2-class field towers for quadratic number fields with 2-class group of type (2,2)

1998 ◽  
Vol 40 (1) ◽  
pp. 63-69 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Field Theory ◽  
Class Number ◽  
Class Group ◽  
Number Fields ◽  
Class Field Theory ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Class Field ◽  
Class Field Tower

Let K be a quadratic number field with 2-class group of type (2,2). Thus if Sk is the Sylow 2-subgroup of the ideal class group of K, then Sk = ℤ/2ℤ × ℤ/2ℤ LetK ⊂ K1 ⊂ K2 ⊂ K3 ⊂…the 2-class field tower of K. Thus K1 is the maximal abelian unramified extension of K of degree a power of 2; K2 is the maximal abelian unramified extension of K of degree a power of 2; etc. By class field theory the Galois group Ga1 (K1/K) ≅ Sk ≅ ℤ/2ℤ × ℤ/2ℤ, and in this case it is known that Ga(K2/Kl) is a cyclic group (cf. [3] and [10]). Then by class field theory the class number of K2 is odd, and hence K2 = K3 = K4 = …. We say that the 2-class field tower of K terminates at K1 if the class number of K1 is odd (and hence K1 = K2 = K3 = … ); otherwise we say that the 2-class field tower of K terminates at K2. Our goal in this paper is to determine how likely it is for the 2-class field tower of K to terminate at K1 and how likely it is for the 2-class field tower of K to terminate at K2. We shall consider separately the imaginary quadratic fields and the real quadratic fields.

On the Hilbert 2-class field of some quadratic number fields

2019 ◽  
Vol 15 (04) ◽  
pp. 807-824 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohammed Rezzougui ◽  
Mohammed Taous ◽  
Abdelkader Zekhnini
Keyword(s):  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Class Field ◽  
Quadratic Number Fields

In this paper, we investigate the cyclicity of the [Formula: see text]-class group of the first Hilbert [Formula: see text]-class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let [Formula: see text] be different prime integers. Put [Formula: see text], and denote by [Formula: see text] its [Formula: see text]-class group and by [Formula: see text] (respectively [Formula: see text]) its first (respectively second) Hilbert [Formula: see text]-class field. Then, we are interested in studying the metacyclicity of [Formula: see text] and the cyclicity of [Formula: see text] whenever the [Formula: see text]-rank of [Formula: see text] is [Formula: see text].

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.

scholarly journals A PRECISE RESULT ON THE ARITHMETIC OF NON-PRINCIPAL ORDERS IN ALGEBRAIC NUMBER FIELDS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250087 ◽  
Author(s):  
ANDREAS PHILIPP
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Theoretical Approach ◽  
Class Group ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Precise Result ◽  
Principal Order

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

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