scholarly journals Simultaneous indivisibility of class numbers of pairs of real quadratic fields

2021 ◽  
Author(s):  
Jaitra Chattopadhyay ◽  
Anupam Saikia
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Fields ◽  
Real Quadratic

scholarly journals On the Insolubility of a Class of Diophantine Equations and the Nontriviality of the Class Numbers of Related Real Quadratic Fields of Richaud-Degert Type

1987 ◽  
Vol 105 ◽  
pp. 39-47 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Diophantine Equations ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Fields ◽  
Integer Solutions ◽  
The Relationship ◽  
Real Quadratic

Many authors have studied the relationship between nontrivial class numbers h(n) of real quadratic fields and the lack of integer solutions for certain diophantine equations. Most such results have pertained to positive square-free integers of the form n = l2 + r with integer >0, integer r dividing 4l and — l<r<l. For n of this form, is said to be of Richaud-Degert (R-D) type (see [3] and [8]; as well as [2], [6], [7], [12] and [13] for extensions and generalizations of R-D types.)

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 Class numbers of real quadratic fields

1998 ◽  
Vol 57 (2) ◽  
pp. 261-274 ◽  
Author(s):  
Jae Moon Kim
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Iwasawa Theory ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
Main Conjecture ◽  
Real Quadratic

Let be a real quadratic field. It is well known that if 3 divides the class number of k, then 3 divides the class number of , and thus it divides B1,χω−1, where χ and ω are characters belonging to the fields k and respectively. In general, the main conjecture of Iwasawa theory implies that if an odd prime p divides the class number of k, then p divides B1,χω−1, where ω is the Teichmüller character for p.The aim of this paper is to examine its converse when p splits in k. Let k∞ be the ℤp-extension of k = k0 and hn be the class number of kn, the n th layer of the ℤp-extension. We shall prove that if p |B1,χω−1, then p | hn for all n ≥ 1. In terms of Iwasawa theory, this amounts to saying that if M∞/k∞, is nontrivial, then L∞/k∞ is nontrivial, where M∞ and L∞ are the maximal abelian p-extensions unramified outside p and unramified everywhere respectively.

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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