scholarly journals Units and2-class field towers of some multiquadratic number fields

2020 ◽  
Vol 44 (4) ◽  
pp. 1466-1483 ◽  
Author(s):  
Mohamed Mahmoud CHEMS-EDDIN ◽  
Abdelkader ZEKHNINI ◽  
Abdelmalek AZIZI
Keyword(s):  
Number Fields ◽  
Class Field ◽  
Class Field Towers

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 INFINITE HILBERT CLASS FIELD TOWERS FROM GALOIS REPRESENTATIONS

2011 ◽  
Vol 07 (01) ◽  
pp. 1-8
Author(s):  
KIRTI JOSHI ◽  
CAMERON MCLEMAN
Keyword(s):  
Galois Representations ◽  
Number Fields ◽  
Modular Representations ◽  
Infinite Class ◽  
Fixed Field ◽  
Class Field ◽  
Class Field Towers ◽  
K 12 ◽  
Hilbert Class Field ◽  
Class Field Tower

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each k ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes ℓ such that the fixed field of the mod-ℓ representation attached to the unique normalized cusp eigenform of weight k on SL2(ℤ) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes ℓ for each k in the list. Finally, given a non-CM curve E/ℚ, we show that there exists an integer ME such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to ME.

On the 2-class field towers of some imaginary quartic cyclic number fields

2019 ◽  
Vol 158 (1) ◽  
pp. 103-118
Author(s):  
Abdelmalek Azizi ◽  
Idriss Jerrari ◽  
Abdelkader Zekhnini ◽  
Mohammed Talbi
Keyword(s):  
Number Fields ◽  
Class Field ◽  
Class Field Towers

scholarly journals Infinite class field towers of number fields of prime power discriminant

2020 ◽  
Vol 373 ◽  
pp. 107318
Author(s):  
Farshid Hajir ◽  
Christian Maire ◽  
Ravi Ramakrishna
Keyword(s):  
Prime Power ◽  
Number Fields ◽  
Infinite Class ◽  
Class Field ◽  
Class Field Towers

scholarly journals Infinite class field towers

2009 ◽  
Vol 344 (4) ◽  
pp. 923-928 ◽  
Author(s):  
Jing Long Hoelscher
Keyword(s):  
Infinite Class ◽  
Class Field ◽  
Class Field Towers
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