On the divisor class number of a real quadratic field. II

1986 ◽  
pp. 7-11
Author(s):  
I. Sh. Slavutskiĭ
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Real Quadratic Field ◽  
Divisor Class ◽  
Real Quadratic

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 Lower Bound for the Class Number of ℚn2+4

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Hasan Sankari ◽  
Ahmad Issa
Keyword(s):  
Lower Bound ◽  
Class Number ◽  
Quadratic Field ◽  
Real Quadratic Field ◽  
Prime Divisors ◽  
Explicit Lower Bound ◽  
Real Quadratic

In this paper, we give an explicit lower bound for the class number of real quadratic field ℚd, where d=n2+4 is a square-free integer, using  ωn which is the number of odd prime divisors of n.

scholarly journals Continued fractions and class number two

2001 ◽  
Vol 27 (9) ◽  
pp. 565-571
Author(s):  
Richard A. Mollin
Keyword(s):  
Continued Fractions ◽  
Class Number ◽  
Quadratic Field ◽  
Ideal Theory ◽  
Quadratic Polynomial ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
New Class ◽  
Real Quadratic

We use the theory of continued fractions in conjunction with ideal theory (often called the infrastructure) in real quadratic fields to give new class number 2 criteria and link this to a canonical norm-induced quadratic polynomial. By doing so, this provides a real quadratic field analogue of the well-known result by Hendy (1974) for complex quadratic fields. We illustrate with several examples.

scholarly journals ON THE IDEAL CLASS GROUP OF CERTAIN QUADRATIC FIELDS

2010 ◽  
Vol 52 (3) ◽  
pp. 575-581 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Ideal Class Group ◽  
The Real ◽  
Real Quadratic ◽  
The Ideal

AbstractLet n(≥ 3) be an odd integer. Let k:= $\Q(\sqrt{4-3^n})\)$ be the imaginary quadratic field and k′:= $\Q(\sqrt{-3(4-3^n)})\)$ the real quadratic field. In this paper, we prove that the class number of k is divisible by 3 unconditionally, and the class number of k′ is divisible by 3 if n(≥ 9) is divisible by 3. Moreover, we prove that the 3-rank of the ideal class group of k is at least 2 if n(≥ 9) is divisible by 3.

On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field

1981 ◽  
Vol 33 (1) ◽  
pp. 55-58 ◽  
Author(s):  
Hiroshi Takeuchi
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Cyclotomic Field ◽  
Rational Numbers ◽  
Real Quadratic Field ◽  
The Real ◽  
Root Of Unity ◽  
Image Position ◽  
Real Quadratic

Let p be an integer and let H(p) be the class-number of the fieldwhere ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).We designate by h(p) the class-number of the real quadratic field

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 ON THE CLASS NUMBER AND THE FUNDAMENTAL UNIT OF THE REAL QUADRATIC FIELD

2012 ◽  
Vol 85 (3) ◽  
pp. 359-370 ◽  
Author(s):  
JAE MOON KIM ◽  
JADO RYU
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Fundamental Unit ◽  
Index Formula ◽  
Real Quadratic Field ◽  
Group Of Units ◽  
Exact Power ◽  
Circular Unit ◽  
Circular Units ◽  
Real Quadratic

AbstractFor a real quadratic field $k=\mathbb {Q}(\sqrt {pq})$, let tk be the exact power of 2 dividing the class number hk of k and ηk the fundamental unit of k. The aim of this paper is to study tk and the value of Nk/ℚ(ηk). Various methods have been successfully applied to obtain results related to this topic. The idea of our work is to select a special circular unit ℰk of k and investigate C(k)=〈±ℰk 〉. We examine the indices [E(k):C(k)] and [C(k):CS (k)] , where E(k) is the group of units of k, and CS (k) is that of circular units of k defined by Sinnott. Then by using the Sinnott’s index formula [E(k):CS (k)]=hk, we obtain as much information about tk and Nk/ℚ (ηk) as possible.

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