Some Results Connected with the Class Number Problem in Real Quadratic Fields

2005 ◽  
Vol 21 (5) ◽  
pp. 1107-1112 ◽  
Author(s):  
Aleksander Grytczuk ◽  
Jarosław Grytczuk
Keyword(s):  
Class Number ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Number Problem ◽  
Real Quadratic

scholarly journals New invariants and class number problem in real quadratic fields

1993 ◽  
Vol 132 ◽  
pp. 175-197 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Problem ◽  
Number Formula ◽  
Positive Integers ◽  
Real Quadratic

In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.

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.

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