scholarly journals Dedekind zeta functions of certain real quadratic fields

2006 ◽  
Vol 37 (4) ◽  
pp. 367-375
Author(s):  
Rongzheng Jiao ◽  
Hongwen Lu
Keyword(s):  
Zeta Functions ◽  
Number Fields ◽  
High Rank ◽  
Quadratic Fields ◽  
Dedekind Sums ◽  
Quadratic Number ◽  
Real Quadratic Fields ◽  
Modular Transformation ◽  
Transformation Methods ◽  
Real Quadratic

Using analytic and modular transformation methods, we represent the value of the product of two Dedekind zeta functions of certain real quadratic number fields at $-3$ by Dedekind sums of high rank in this paper.

EXPONENTS OF CLASS GROUPS OF REAL QUADRATIC FIELDS

2008 ◽  
Vol 04 (04) ◽  
pp. 597-611 ◽  
Author(s):  
KALYAN CHAKRABORTY ◽  
FLORIAN LUCA ◽  
ANIRBAN MUKHOPADHYAY
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Quadratic Number ◽  
Class Groups ◽  
Real Quadratic Fields ◽  
Prime Factors ◽  
Positive Integers ◽  
Real Quadratic

In this paper, we show that the number of real quadratic fields 𝕂 of discriminant Δ𝕂 < x whose class group has an element of order g (with g even) is ≥ x1/g/5 if x > x0, uniformly for positive integers g ≤ ( log log x)/(8 log log log x). We also apply the result to find real quadratic number fields whose class numbers have many prime factors.

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.

ELEMENTS OF ORDER FOUR IN THE NARROW CLASS GROUP OF REAL QUADRATIC FIELDS

2015 ◽  
Vol 100 (1) ◽  
pp. 21-32
Author(s):  
ELLIOT BENJAMIN ◽  
C. SNYDER
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Gaussian Integers ◽  
Real Quadratic Fields ◽  
Order Four ◽  
Real Quadratic ◽  
Narrow Class

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

scholarly journals New number fields with known p-class tower

2015 ◽  
Vol 64 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Finite Index ◽  
Number Fields ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Abstract The p-class tower F∞p (k) of a number field k is its maximal unramified pro-p extension. It is considered to be known when the p-tower group, that is the Galois group G := Gal(F∞p (k)|k), can be identified by an explicit presentation. The main intention of this article is to characterize assigned finite 3-groups uniquely by abelian quotient invariants of subgroups of finite index, and to provide evidence of actual realizations of these groups by 3-tower groups G of real quadratic fields K = ℚ( √d) with 3-capitulation type (0122) or (2034).

Sign in / Sign up

Export Citation Format

Share Document