Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields by Integer Lattice Reduction

Number Theory ◽  
1991 ◽  
pp. 149-202 ◽  
Author(s):  
Erich Kaltofen ◽  
Noriko Yui
Keyword(s):  
Explicit Construction ◽  
Integer Lattice ◽  
Lattice Reduction ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Class Fields

Completely normal elements in some finite abelian extensions

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Ja Koo ◽  
Dong Shin
Keyword(s):  
Function Fields ◽  
Modular Function ◽  
Rational Numbers ◽  
Quadratic Fields ◽  
Cyclotomic Fields ◽  
Imaginary Quadratic Fields ◽  
Abelian Extensions ◽  
Class Fields ◽  
Normal Element ◽  
Ray Class Fields

AbstractWe present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

Construction of ray-class fields by smaller generators and their applications

2017 ◽  
Vol 147 (4) ◽  
pp. 781-812 ◽  
Author(s):  
Ja Kyung Koo ◽  
Dong Sung Yoon
Keyword(s):  
Diophantine Equations ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Minimal Polynomials ◽  
Class Invariants ◽  
Class Fields ◽  
Ray Class Fields

We generate ray-class fields over imaginary quadratic fields in terms of Siegel–Ramachandra invariants, which are an extension of a result of Schertz. By making use of quotients of Siegel–Ramachandra invariants we also construct ray-class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

scholarly journals Lattice Reduction Over Imaginary Quadratic Fields

2020 ◽  
Vol 68 ◽  
pp. 6380-6393
Author(s):  
Shanxiang Lyu ◽  
Christian Porter ◽  
Cong Ling
Keyword(s):  
Lattice Reduction ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields

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.

scholarly journals On some Fricke families and application to the Lang–Schertz conjecture

2016 ◽  
Vol 146 (4) ◽  
pp. 723-740 ◽  
Author(s):  
Ho Yun Jung ◽  
Ja Kyung Koo ◽  
Dong Hwa Shin
Keyword(s):  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Special Values ◽  
Class Fields ◽  
Ray Class Fields

We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang–Schertz conjecture.

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