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 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.

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