scholarly journals Construction of 3-Hilbert class field of certain imaginary quadratic fields

2010 ◽  
Vol 86 (1) ◽  
pp. 18-19 ◽  
Author(s):  
Jangheon Oh
Keyword(s):  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Class Field ◽  
Hilbert Class Field

scholarly journals Computing the Hilbert class field of real quadratic fields

1999 ◽  
Vol 69 (231) ◽  
pp. 1229-1245 ◽  
Author(s):  
Henri Cohen ◽  
Xavier-François Roblot
Keyword(s):  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Hilbert Class Field ◽  
Real Quadratic

Class field towers of imaginary quadratic fields

1987 ◽  
Vol 57 (4) ◽  
pp. 425-450 ◽  
Author(s):  
James R. Brink ◽  
Robert Gold
Keyword(s):  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Class Field ◽  
Class Field Towers

scholarly journals The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields

2015 ◽  
Vol 11 (06) ◽  
pp. 1961-2017 ◽  
Author(s):  
Rodney Lynch ◽  
Patrick Morton
Keyword(s):  
Nontrivial Solution ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Explicit Formulas ◽  
Odd Degree ◽  
Class Fields ◽  
Fermat Equation ◽  
Hilbert Class Field ◽  
Integral Solutions

It is shown that the quartic Fermat equation x4 + y4 = 1 has nontrivial integral solutions in the Hilbert class field Σ of any quadratic field [Formula: see text] whose discriminant satisfies -d ≡ 1 (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in [Formula: see text], for p (> 7) a prime congruent to 7 (mod 8), but does have a nontrivial solution in the odd degree extension Σ of K. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in detail and the results are applied to prove several properties of the Weber singular moduli introduced by Yui and Zagier.

Special units and ideal class groups of extensions of imaginary quadratic fields

2007 ◽  
Vol 143 (2) ◽  
pp. 265-270
Author(s):  
BYUNGCHUL CHA
Keyword(s):  
Quadratic Field ◽  
Abelian Extension ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Imaginary Quadratic Fields ◽  
Class Field ◽  
Ideal Class Groups ◽  
Module Structures ◽  
Hilbert Class Field ◽  
The Ideal

Let K be an imaginary quadratic field, and let F be an abelian extension of K, containing the Hilbert class field of K. We fix a rational prime p > 2 which does not divide the number of roots of unity in the Hilbert class field of K. Also, we assume that the prime p does not divide the order of the Galois group G:=Gal(F/K). Let AF be the ideal class group of F, and EF be the group of global units of F. The purpose of this paper is to study the Galois module structures of AF and EF.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

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