scholarly journals Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

2012 ◽  
Vol 2012 ◽  
pp. 1-4
Author(s):  
A. Pekin
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Sufficient Condition ◽  
Prime Divisors ◽  
Imaginary Quadratic Fields ◽  
Prime Factors

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2g, whereg>1is an integer and the discriminant of such fields has only two prime divisors.

scholarly journals On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

2021 ◽  
Vol 197 (1) ◽  
pp. 105-110
Author(s):  
Jaitra Chattopadhyay ◽  
Subramani Muthukrishnan
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields

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 REMARKS ON THE DIVISIBILITY OF THE CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS

2011 ◽  
Vol 53 (2) ◽  
pp. 379-389 ◽  
Author(s):  
AKIKO ITO
Keyword(s):  
Prime Number ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe consider the divisibility of the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$, where q is an odd prime number, k and n are positive integers. Suppose that k ≡ 1 mod 2 or n ≢ 3 mod 6. We show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 3 mod 8. This is a generalization of the result of Kishi for imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - 3^n})$ when k ≡ 1 mod 2 or n ≢ 3 mod 6. We also show that the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-1})$ are divisible by n for q ≡ 1 mod 4 and the class numbers of imaginary quadratic fields $\mathbb{Q}(\sqrt{2^{2k} - q^n})$ ≠ $\mathbb{Q}(\sqrt{-3})$ are divisible by n for q ≡ 7 mod 8.

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