A positive proportion of some quadratic number fields with infinite Hilbert 2-class field tower

2015 ◽  
Vol 40 (2) ◽  
pp. 405-412
Author(s):  
A. Mouhib
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Class Field ◽  
Positive Proportion ◽  
Quadratic Number Fields ◽  
Class Field Tower

The narrow 2-class field tower of some real quadratic number fields

2020 ◽  
Vol 194 (2) ◽  
pp. 187-217
Author(s):  
Elliot Benjamin ◽  
C. Snyder
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Class Field ◽  
Quadratic Number Fields ◽  
Class Field Tower ◽  
Real Quadratic

scholarly journals Real quadratic number fields with metacyclic Hilbert $2$-class field tower

2018 ◽  
Vol 144 (2) ◽  
pp. 177-190
Author(s):  
Said Essahel ◽  
Ahmed Dakkak ◽  
Ali Mouhib
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Class Field ◽  
Quadratic Number Fields ◽  
Class Field Tower ◽  
Real Quadratic

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.

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