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

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.

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

scholarly journals On the Doi-Naganuma lifting associated with imaginary quadratic fields

1978 ◽  
Vol 71 ◽  
pp. 149-167 ◽  
Author(s):  
Tetsuya Asai
Keyword(s):  
Theta Function ◽  
Function Method ◽  
Quadratic Field ◽  
Discrete Subgroup ◽  
Real Quadratic Field ◽  
Field Case ◽  
The Real ◽  
Concrete Form ◽  
Elliptic Modular Form ◽  
Real Quadratic

Similarly to the real quadratic field case by Doi and Naganuma ([3], [9]) there is a lifting from an elliptic modular form to an automorphic form on SL2(C) with respect to an arithmetic discrete subgroup relative to an imaginary quadratic field. This fact is contained in his general theory of Jacquet ([6]) as a special case. In this paper, we try to reproduce this lifting in its concrete form by using the theta function method developed first by Niwa ([10]); also Kudla ([7]) has treated the real quadratic field case on the same line. The theta function method will naturally lead to a theory of lifting to an orthogonal group of general signature (cf. Oda [11]), and the present note will give a prototype of non-holomorphic case.

scholarly journals On Petersson’s partition limit formula

2020 ◽  
pp. 1-14
Author(s):  
Carlos Castaño-Bernard ◽  
Florian Luca
Keyword(s):  
Quadratic Field ◽  
Character Formula ◽  
Real Quadratic Field ◽  
The Real ◽  
Limit Formula ◽  
Real Quadratic

For each prime [Formula: see text] consider the Legendre character [Formula: see text]. Let [Formula: see text] be the number of partitions of [Formula: see text] into parts [Formula: see text] such that [Formula: see text]. Petersson proved a beautiful limit formula for the ratio of [Formula: see text] to [Formula: see text] as [Formula: see text] expressed in terms of important invariants of the real quadratic field [Formula: see text]. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz–Cesàro theorem. In this paper, we suggest an approach to address Grosswald’s conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman–Erdős.

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

Properties of real quadratic irrational numbers under the action of the group H = áx, y: x2 = y4 = 1ñ

2005 ◽  
Vol 42 (4) ◽  
pp. 371-386
Author(s):  
M. Aslam Malik ◽  
S. M. Husnine ◽  
Abdul Majeed
Keyword(s):  
Quadratic Field ◽  
Infinite Group ◽  
Algebraic Structures ◽  
Real Quadratic Field ◽  
Irrational Numbers ◽  
The Real ◽  
Basic Properties ◽  
Quadratic Irrational ◽  
Real Quadratic

Studying groups through their actions on different sets and algebraic structures has become a useful technique to know about the structure of the groups. The main object of this work is to examine the action of the infinite group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H = \langle x,y : x^{2} = y^{4} = 1\rangle$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $x (z) = \frac{-1}{2z}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $y (z) = \frac{-1}{2(z+1)}$ \end{document} on the real quadratic field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} and find invariant subsets of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H$ \end{document}. We also discuss some basic properties of elements of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group H.

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 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].

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