scholarly journals Imaginary bicyclic biquadratic fields with the real quadratic subfield of class-number one

1986 ◽  
Vol 102 ◽  
pp. 91-100
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
The Other ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
The Real ◽  
Real Quadratic Fields ◽  
Other Hand ◽  
Biquadratic Fields ◽  
Real Quadratic

It has been proved by A. Baker [1] and H. M. Stark [7] that there exist exactly 9 imaginary quadratic fields of class-number one. On the other hand, G.F. Gauss has conjectured that there exist infinitely many real quadratic fields of class-number one, and the conjecture is now still unsolved.

scholarly journals The fundamental unit and bounds for class numbers of real quadratic fields

1991 ◽  
Vol 124 ◽  
pp. 181-197 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Although class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker [3] and by H. M. Stark [25] independently, the problem for real quadratic fields remains still unsettled. However, since papers by Ankeny–Chowla–Hasse [2] and H. Hasse [9], many papers concerning this problem or giving estimate for class numbers of real quadratic fields from below have appeared. There are three methods used there, namely the first is related with quadratic diophantine equations ([2], [9], [27, 28, 29, 31], [17]), and the second is related with continued fraction expantions ([8], [4], [16], [14], [18]).

scholarly journals The fundamental unit and class number one problem of real quadratic fields with prime discriminant

1990 ◽  
Vol 120 ◽  
pp. 51-59 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker and H.M. Stark independently. However, the problem for real quadratic fields is still unsolved. It seems to us that one of the most essential difficulties of the problem for real quadratic fields comes from deep connection of the class number with the fundamental unit.

scholarly journals On determining certain real quadratic fields with class number one and relating this property to continued fractions and primality properties

1991 ◽  
Vol 124 ◽  
pp. 157-180 ◽  
Author(s):  
Eugène Dubois ◽  
Claude Levesque
Keyword(s):  
Continued Fractions ◽  
Class Number ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Thanks to K. Heegner [He], A. Baker [Ba] and H. Stark [S], we know that there are nine imaginary quadratic fields of class number one. Gauss conjectured that there are infinitely many real quadratic fields of class number one, but the conjecture is still open.

LE 2-RANG DU GROUPE DE CLASSES DE CERTAINS CORPS BIQUADRATIQUES ET APPLICATIONS

2004 ◽  
Vol 15 (02) ◽  
pp. 169-182
Author(s):  
ABDELMALEK AZIZI ◽  
ALI MOUHIB
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Class Field ◽  
Real Quadratic Fields ◽  
Biquadratic Fields ◽  
Class Field Tower ◽  
Real Quadratic

Let K be a real biquadratic field and let k be a quadratic field with odd class number contained in K. The aim of this article is to determine the rank of the 2-class group of K and we give applications to the structure of the 2-class group of some biquadratic fields and to the 2-class field tower of some real quadratic fields. Résumé: Soient K un corps biquadratique réel et k un sous-corps quadratique de K dont le nombre de classes est impair. Dans ce papier on détermine le rang du 2-groupe de classes de K et on donne des applications à la structure du 2-groupe de classes de certains corps biquadratiques et aussi à la tour des 2-corps de classes de Hilbert de certains corps quadratiques réels.

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