Congruence relations of Ankeny-Artin-Chowla type for pure cubic fields

Ankeny, Artin and Chowla [1] proved a congruence relation among the class number, the fundamental unit of real quadratic fields, and the Bernoulli numbers. Our aim of this paper is to prove similar congruence relations for pure cubic fields. For this purpose, we use the Hurwitz numbers associated with the elliptic curve defined by y2 = 4x3 — 1 instead of the Bernoulli numbers (§ 3).

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Congruences of Ankeny-Artin-Chowla type for pure quartic and sectic fields

Nagoya Mathematical Journal ◽

10.1017/s0027763000002695 ◽

1987 ◽

Vol 108 ◽

pp. 131-144

Author(s):

Masato Kamei

Keyword(s):

Bernoulli Numbers ◽

Quadratic Fields ◽

Class Numbers ◽

Exact Form ◽

Hurwitz Numbers ◽

Real Quadratic Fields ◽

Cubic Fields ◽

Real Quadratic

Ankeny, Artin and Chowla [1] showed that there are congruences between class numbers of real quadratic fields and generalized Bernoulli numbers. Recently, Ito [3] has extended their results to the case of pure cubic fields using generalized Hurwitz numbers of Lichtenbaum [4]. In his paper, he suggested that similar results would be obtained for pure quartic and sectic fields. In this paper, we carry out this by following his idea. To give a congruence in an exact form, we need an idea due to Matthews [5]. As the argument in the sectic case is quite parallel to that in the quartic case, we shall discuss the former case briefly in the last two sections.

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On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

International Mathematics Research Notices ◽

10.1093/imrn/rnaa326 ◽

2020 ◽

Author(s):

Joachim Petit

Keyword(s):

Asymptotic Formula ◽

Elliptic Curve ◽

Asymptotic Estimate ◽

Rational Point ◽

Quadratic Fields ◽

Real Quadratic Fields ◽

Quadratic Twists ◽

The Family ◽

Minimal Height ◽

Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

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The Gauss-Hasse hypothesis on real quadratic fields with class number one

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1982.335.40 ◽

1982 ◽

Vol 1982 (335) ◽

pp. 40-86

Keyword(s):

Class Number ◽

Quadratic Fields ◽

Real Quadratic Fields ◽

Real Quadratic

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The Class-Number of Real Quadratic Fields

Springer Collected Works in Mathematics - Collected Papers ◽

10.1007/978-1-4614-6294-1_15 ◽

1965 ◽

pp. 232-233

Author(s):

Emil Artin

Keyword(s):

Class Number ◽

Quadratic Fields ◽

Real Quadratic Fields ◽

Real Quadratic

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The Class-Number of Real Quadratic Fields

Emil Artin Collected Papers ◽

10.1007/978-1-4612-5717-2_16 ◽

1965 ◽

pp. 232-233

Author(s):

N. C. Ankeney ◽

E. Artin ◽

S. Chowla

Keyword(s):

Class Number ◽

Quadratic Fields ◽

Real Quadratic Fields ◽

Real Quadratic

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Prime producing quadratic polynomials and class-number one problem for real quadratic fields

Proceedings of the Japan Academy Series A Mathematical Sciences ◽

10.3792/pjaa.66.119 ◽

1990 ◽

Vol 66 (5) ◽

pp. 119-121 ◽

Cited By ~ 3

Author(s):

Masaki Kobayashi

Keyword(s):

Class Number ◽

Quadratic Fields ◽

Quadratic Polynomials ◽

Real Quadratic Fields ◽

Real Quadratic

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Class number one problem for real quadratic fields (the conjecture of Gauss)

Proceedings of the Japan Academy Series A Mathematical Sciences ◽

10.3792/pjaa.64.53 ◽

1988 ◽

Vol 64 (2) ◽

pp. 53-55 ◽

Cited By ~ 4

Author(s):

Hideo Yokoi

Keyword(s):

Class Number ◽

Quadratic Fields ◽

Real Quadratic Fields ◽

Real Quadratic

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A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2019.2.00033 ◽

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.

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