scholarly journals Class number one problem for real quadratic fields (the conjecture of Gauss)

1988 ◽  
Vol 64 (2) ◽  
pp. 53-55 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

The Class-Number of Real Quadratic Fields

1965 ◽  
pp. 232-233
Author(s):  
N. C. Ankeney ◽  
E. Artin ◽  
S. Chowla
Keyword(s):  
Class Number ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

scholarly journals A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

2019 ◽  
Vol 5 (1) ◽  
pp. 495-498
Author(s):  
Özen Özer
Keyword(s):  
Number Theory ◽  
Class Number ◽  
Number Fields ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Integral Basis ◽  
Real Quadratic Fields ◽  
Class Number Formula ◽  
Number Formula ◽  
Real Quadratic

AbstractDifferent types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛd of {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ-function and fundamental unit ɛd are significant and necessary tools for determining the structure of real quadratic fields.The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element wd, fundamental unit ɛd, and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

Another look at real quadratic fields of relative class number 1

2014 ◽  
Vol 163 (4) ◽  
pp. 371-377 ◽  
Author(s):  
Debopam Chakraborty ◽  
Anupam Saikia
Keyword(s):  
Class Number ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Relative Class ◽  
Relative Class Number ◽  
Real Quadratic

On the unit group and the class number¶of certain composita of two real quadratic fields

2001 ◽  
Vol 105 (1) ◽  
pp. 85-101
Author(s):  
Shin-ichi Katayama ◽  
Claude Levesque ◽  
Toru Nakahara
Keyword(s):  
Class Number ◽  
Unit Group ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic
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