scholarly journals A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge Garcia Villeda
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Dirichlet Character ◽  
Imaginary Quadratic Field ◽  
Kronecker Symbol ◽  
Quadratic Residues ◽  
Elementary Methods ◽  
Computable Formula

<p style='text-indent:20px;'>Using elementary methods, we count the quadratic residues of a prime number of the form <inline-formula><tex-math id="M2">\begin{document}$ p = 4n-1 $\end{document}</tex-math></inline-formula> in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number <inline-formula><tex-math id="M3">\begin{document}$ h $\end{document}</tex-math></inline-formula> of the imaginary quadratic field <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Q(\sqrt{-p}). $\end{document}</tex-math></inline-formula> Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.</p>

A certain eta-quotient and the class number of an imaginary quadratic field

2014 ◽  
Vol 10 (06) ◽  
pp. 1485-1499
Author(s):  
Takeshi Ogasawara
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Theta Series ◽  
Imaginary Quadratic Field

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field.

Congruence of Ankeny–Artin–Chowla type for cyclic fields of prime degree l

1996 ◽  
Vol 119 (1) ◽  
pp. 17-22 ◽  
Author(s):  
Stanislav Jakubec
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Fundamental Unit ◽  
Prime Degree ◽  
Image Position

Ankeny–Artin–Chowla obtained several congruences for the class number hk of a quadratic field K, some of which were also obtained by Kiselev. In particular, if the discriminant of K is a prime number p ≡ 1 (mod 4) and ε = t + u √p/2 is the fundamental unit of K, then

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

scholarly journals Class Numbers and Biquadratic Reciprocity

1982 ◽  
Vol 34 (4) ◽  
pp. 969-988 ◽  
Author(s):  
Kenneth S. Williams ◽  
James D. Currie
Keyword(s):  
Real Number ◽  
Class Number ◽  
Quadratic Field ◽  
Class Numbers ◽  
Kronecker Symbol ◽  
The Real ◽  
Unique Manner ◽  
Image Position

0. Notation. Throughout this paper p denotes a prime congruent to 1 modulo 4. It is well known that such primes are expressible in an essentially unique manner as the sum of the squares of two integers, that is,(0.1)with |a| and |b| uniquely determined by (0.1). Since a is odd, replacing a by –a if necessary, we can specify a uniquely by(0.2)Further, as {[(p – l)/2]!}2 = – 1 (mod p), we can specify b uniquely by(0.3)These choices are assumed throughout.The following notation is also used throughout the paper: h(d) denotes the class number of the quadratic field of discriminant d, (d/n) is the Kronecker symbol of modulus |d|, [x] denotes the greatest integer less than or equal to the real number x, and {x} = x – [x].

scholarly journals THE DIVISIBILITY OF THE CLASS NUMBER OF THE IMAGINARY QUADRATIC FIELD

2011 ◽  
Vol 54 (1) ◽  
pp. 149-154 ◽  
Author(s):  
ZHU MINHUI ◽  
WANG TINGTING
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Positive Integers

AbstractLet hK denote the class number of the imaginary quadratic field $K=\mathbf{Q}(\sqrt{2^{2m}-k^n})$, where m and n are positive integers, k is an odd integer with k > 1 and 22m < kn. In this paper we prove that if either 3 ∣ n and 22m − kn ≡ 5(mod 8) or n = 3 and k = (22m+2 −1)/3, then ∣ hK. Otherwise, we have n ∣ hK.

scholarly journals Limiting equations for the class number of the ideals of an imaginary quadratic field

1965 ◽  
Vol 5 (2) ◽  
pp. 303-305
Author(s):  
O. Saparnijazov
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: О. Сапарниязов. Асимптотические равенства для числа классов идеалов мнимого квадратического поля O. Saparnijazovas. Menamojo kvadratinio skaičių kūno idealų klasių skaičiaus asimptotinės išraiškos

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

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