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.

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.

scholarly journals On the class number formula of certain real quadratic fields

2013 ◽  
Vol Volume 36 ◽  
Author(s):  
K Chakraborty ◽  
S Kanemitsu ◽  
T Kuzumaki
Keyword(s):  
Class Number ◽  
Character Sums ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
International Audience ◽  
Special Case ◽  
Real Quadratic

International audience In this note we give an alternate expression of class number formula for real quadratic fields with discriminant $d \equiv 5\, {\rm mod}\, 8$. %Dirichlet's classical class number formula for real quadratic fields expresses `class number' in somewhat `transcend' manner, which was simplified by P. Chowla in the special case when the discriminant $d = p \equiv 5\,{\rm mod}\, 8$ is a prime. We use another form of class number formula and transform it using Dirichlet's $1/4$-th character sums. Our result elucidates other generalizations of the class number formula by Mitsuhiro, Nakahara and Uhera for general real quadratic fields.

The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit

1995 ◽  
Vol 38 (1) ◽  
pp. 98-103
Author(s):  
T. Mitsuhiro ◽  
T. Nakahara ◽  
T. Uehara
Keyword(s):  
Analytical Method ◽  
Class Number ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
Class Number Formula ◽  
Number Formula ◽  
Real Quadratic

AbstractOur aim is to give an arithmetical expression of the class number formula of real quadratic fields. Starting from the classical Dirichlet class number formula, our proof goes along arithmetical lines not depending on any analytical method such as an estimate for

scholarly journals AN ANALOGUE OF CIRCULAR UNITS FOR PRODUCTS OF ELLIPTIC CURVES

2004 ◽  
Vol 47 (1) ◽  
pp. 35-51 ◽  
Author(s):  
Srinath Baba ◽  
Ramesh Sreekantan
Keyword(s):  
Elliptic Curves ◽  
Class Number ◽  
Quadratic Fields ◽  
Motivic Cohomology ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
Class Number Formula ◽  
Number Formula ◽  
Circular Units ◽  
Real Quadratic

AbstractWe construct certain elements in the motivic cohomology group $H^3_{\mathcal{M}}(E\times E',\mathbb{Q}(2))$, where $E$ and $E'$ are elliptic curves over $\mathbb{Q}$. When $E$ is not isogenous to $E'$ these elements are analogous to circular units in real quadratic fields, as they come from modular parametrizations of the elliptic curves. We then find an analogue of the class-number formula for real quadratic fields, which specializes to the usual quadratic class-number formula when $E$ and $E'$ are quadratic twists.AMS 2000 Mathematics subject classification: Primary 11F67; 14G35. Secondary 11F11; 11E45; 14G10

scholarly journals On a class number formula for real quadratic number fields

2002 ◽  
Vol 65 (2) ◽  
pp. 259-270 ◽  
Author(s):  
David M. Bradley ◽  
Ali E. Özlük ◽  
C. Snyder
Keyword(s):  
Class Number ◽  
Dirichlet Character ◽  
Number Fields ◽  
Binomial Coefficients ◽  
Quadratic Number ◽  
Numerical Series ◽  
Class Number Formula ◽  
Quadratic Number Fields ◽  
Number Formula ◽  
Real Quadratic

For an even Dirichlet character ψ, we obtain a formula for L (1, ψ) in terms of a sum of Dirichlet L-Series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coefficients. We then derive a class number formula for real quadratic number fields by taking L (s, ψ) to be the quadratic L-series associated with these fields.

scholarly journals On prime valued polynomials and class numbers of real quadratic fields

1988 ◽  
Vol 112 ◽  
pp. 143-151 ◽  
Author(s):  
R.A. Mollin ◽  
H.C. Williams
Keyword(s):  
Number Theory ◽  
Open Problem ◽  
Class Number ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Fields ◽  
The Moment ◽  
Real Quadratic

Gauss conjectured that there are infinitely many real quadratic fields with class number one. Today this is still an open problem. Moreover, as Dorian Goldfeld, one of the recipients of the 1987 Cole prize in number theory (for his work on another problem going back to Gauss) recently stated in his acceptance of the award: “This problem appears quite intractible at the moment.” However there has recently been a search for conditions which are tantamount to class number one for real quadratic fields. This may be viewed as an effort to shift the focus of the problem in order to understand more clearly the inherent difficulties, and to reveal some other beautiful interrelationships.

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