scholarly journals On imaginary quadratic number fields with 2-class group of rank 4 and infinite 2-class field tower

2001 ◽  
Vol 201 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Elliot Benjamin
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Quadratic Number ◽  
Class Field ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Class Field Tower

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.

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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