Binary Quadratic Forms and the Ideal Class Group
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EVERY ABELIAN GROUP IS the CLASS GROUP of a SIMPLE DEDEKIND DOMAIN 2 Class Group
EVERY ABELIAN GROUP IS THE CLASS GROUP OF A SIMPLE DEDEKIND DOMAIN DANIEL SMERTNIG Abstract. A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind domains form a second important class. We show that every abelian group is the class group of a noncommutative simple Dedekind domain. This solves an open problem stated by Levy and Robson in their recent monograph on hereditary Noetherian prime rings. 1. Introduction Throughout the paper, a domain is a not necessarily commutative unital ring in which the zero element is the unique zero divisor. In [Cla66], Claborn showed that every abelian group G is the class group of a commutative Dedekind domain. An exposition is contained in [Fos73, Chapter III §14]. Similar existence results, yield- ing commutative Dedekind domains which are more geometric, respectively num- ber theoretic, in nature, were obtained by Leedham-Green in [LG72] and Rosen in [Ros73, Ros76]. Recently, Clark in [Cla09] showed that every abelian group is the class group of an elliptic commutative Dedekind domain, and that this domain can be chosen to be the integral closure of a PID in a quadratic field extension. See Clark’s article for an overview of his and earlier results. In commutative mul- tiplicative ideal theory also the distribution of nonzero prime ideals within the ideal classes plays an important role. For an overview of realization results in this direction see [GHK06, Chapter 3.7c]. A ring R is a Dedekind prime ring if every nonzero submodule of a (left or right) progenerator is a progenerator (see [MR01, Chapter 5]). -
Quadratic Forms, Lattices, and Ideal Classes
Quadratic forms, lattices, and ideal classes Katherine E. Stange March 1, 2021 1 Introduction These notes are meant to be a self-contained, modern, simple and concise treat- ment of the very classical correspondence between quadratic forms and ideal classes. In my personal mental landscape, this correspondence is most naturally mediated by the study of complex lattices. I think taking this perspective breaks the equivalence between forms and ideal classes into discrete steps each of which is satisfyingly inevitable. These notes follow no particular treatment from the literature. But it may perhaps be more accurate to say that they follow all of them, because I am repeating a story so well-worn as to be pervasive in modern number theory, and nowdays absorbed osmotically. These notes require a familiarity with the basic number theory of quadratic fields, including the ring of integers, ideal class group, and discriminant. I leave out some details that can easily be verified by the reader. A much fuller treatment can be found in Cox's book Primes of the form x2 + ny2. 2 Moduli of lattices We introduce the upper half plane and show that, under the quotient by a natural SL(2; Z) action, it can be interpreted as the moduli space of complex lattices. The upper half plane is defined as the `upper' half of the complex plane, namely h = fx + iy : y > 0g ⊆ C: If τ 2 h, we interpret it as a complex lattice Λτ := Z+τZ ⊆ C. Two complex lattices Λ and Λ0 are said to be homothetic if one is obtained from the other by scaling by a complex number (geometrically, rotation and dilation). -
Quadratic Form - Wikipedia, the Free Encyclopedia
Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form Quadratic form From Wikipedia, the free encyclopedia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). Contents 1 Introduction 2 History 3 Real quadratic forms 4 Definitions 4.1 Quadratic spaces 4.2 Further definitions 5 Equivalence of forms 6 Geometric meaning 7 Integral quadratic forms 7.1 Historical use 7.2 Universal quadratic forms 8 See also 9 Notes 10 References 11 External links Introduction Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: where a,…,f are the coefficients.[1] Note that quadratic functions, such as ax2 + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the 1 of 8 27/03/2013 12:41 Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. -
Class Group Computations in Number Fields and Applications to Cryptology Alexandre Gélin
Class group computations in number fields and applications to cryptology Alexandre Gélin To cite this version: Alexandre Gélin. Class group computations in number fields and applications to cryptology. Data Structures and Algorithms [cs.DS]. Université Pierre et Marie Curie - Paris VI, 2017. English. NNT : 2017PA066398. tel-01696470v2 HAL Id: tel-01696470 https://tel.archives-ouvertes.fr/tel-01696470v2 Submitted on 29 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Spécialité Informatique École Doctorale Informatique, Télécommunications et Électronique (Paris) Présentée par Alexandre GÉLIN Pour obtenir le grade de DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE Calcul de Groupes de Classes d’un Corps de Nombres et Applications à la Cryptologie Thèse dirigée par Antoine JOUX et Arjen LENSTRA soutenue le vendredi 22 septembre 2017 après avis des rapporteurs : M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern devant le jury composé de : M. Karim BELABAS Professeur, Université de Bordeaux M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern M. -
Math 154. Class Groups for Imaginary Quadratic Fields
Math 154. Class groups for imaginary quadratic fields Homework 6 introduces the notion of class group of a Dedekind domain; for a number field K we speak of the \class group" of K when we really mean that of OK . An important theorem later in the course for rings of integers of number fields is that their class group is finite; the size is then called the class number of the number field. In general it is a non-trivial problem to determine the class number of a number field, let alone the structure of its class group. However, in the special case of imaginary quadratic fields there is a very explicit algorithm that determines the class group. The main point is that if K is an imaginary quadratic field with discriminant D < 0 and we choose an orientation of the Z-module OK (or more concretely, we choose a square root of D in OK ) then this choice gives rise to a natural bijection between the class group of K and 2 2 the set SD of SL2(Z)-equivalence classes of positive-definite binary quadratic forms q(x; y) = ax + bxy + cy over Z with discriminant 4ac − b2 equal to −D. Gauss developed \reduction theory" for binary (2-variable) and ternary (3-variable) quadratic forms over Z, and via this theory he proved that the set of such forms q with 1 ≤ a ≤ c and jbj ≤ a (and b ≥ 0 if either a = c or jbj = a) is a set of representatives for the equivalence classes in SD. -
On a Special Class of Dedekind Domains
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topobgy Vol. 3. Suppl. I, pp. 113-118. Pergamon Press. 1964. Printed in Great Britain ON A SPECIAL CLASS OF DEDEKIND DOMAINS OSCARGOLDMAN (Received 17 May 1963) $1. INTRODUCTION WE ARE interested in Dedekind domains R which have the following two properties: Fl : R/‘j3 is a finite field for every maximal ideal Q F2: The group U(R) of units of R is finitely generated. The ring of integers of an algebraic number field and the Dedekind domains associated to curves over finite fields are examples of rings having the properties listed above. It is the main purpose of this note to show that there are many other examples and that, in fact, the quotient field of such a ring may have arbitrary, but finite, transcendence degree over the prime field. The particular manner in which these domains are constructed (93) suggests that they may have some geometric significance; but that interpretation must be left for another occasion. Certain elementary facts about the relation between units and ideal classes of a Dede- kind domain seem not to have appeared in the literature. Since we shall need to use some of these facts, we shall include them here. $2. UNITS AND IDEAL CLASSES We begin by recalling some well-known facts. If R is a Dedekind domain with quotient field K and ‘$ is a maximal ideal of R, then R, is a discrete rank one valuation ring and R is the intersection of all these R,. -
Integral Binary Quadratic Forms
INTEGRAL BINARY QUADRATIC FORMS UTAH SUMMER REU JUNE JUNE 2 2 An integral binary quadratic form is a p olynomial ax bxy cy where a b and 2 2 c are integers One of the simplest quadratic forms that comes to mind is x y A classical problem in numb er theory is to describ e which integers are sums of two squares More generally one can ask what are integral values of any quadratic form The aim of this workshop is to intro duce binary quadratic forms following a b o ok The Sensual Quadratic Form by Conway and then go on to establish some deep prop erties such as the Gauss group law and a connection with continued fractions More precisely we intend to cover the following topics Integral values In his b o ok Conway develops a metho d to visualize values of a quadratic binary form by intro ducing certain top ographic ob jects rivers lakes etc Gauss law In Gauss dened a comp osition law of binary quadratic forms 2 of a xed discriminant D b ac In a mo dern language this group law corresp onds to multiplication of ideals in the quadratic eld of discriminant D We shall take an approach to this topic from Manjul Bhargavas Ph D thesis where the Gauss law is dened using integer cub es Reduction theory Two quadratic forms can dier only by a change of co ordi nates Such quadratic forms are called equivalent We shall show that there are only nitely many equivalence classes of binary quadratic forms of a xed discriminant D If D the equivalence classes can b e interpreted as cycles of purely p erio dic continued fractions This interpretation -
Ideals and Class Groups of Number Fields
Ideals and class groups of number fields A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science by Minjiao Yang August, 2018 ○C Copyright All rights reserved Except for previously published materials Thesis written by Minjiao Yang B.S., Kent State University, 2015 M.S., Kent State University, 2018 Approved by Gang Yu , Advisor Andrew Tonge , Chair, Department of Mathematics Science James L. Blank , Dean, College of Arts and Science TABLE OF CONTENTS…………………………………………………………...….…...iii ACKNOWLEDGEMENTS…………………………………………………………....…...iv CHAPTER I. Introduction…………………………………………………………………......1 II. Algebraic Numbers and Integers……………………………………………......3 III. Rings of Integers…………………………………………………………….......9 Some basic properties…………………………………………………………...9 Factorization of algebraic integers and the unit group………………….……....13 Quadratic integers……………………………………………………………….17 IV. Ideals……..………………………………………………………………….......21 A review of ideals of commutative……………………………………………...21 Ideal theory of integer ring 풪푘…………………………………………………..22 V. Ideal class group and class number………………………………………….......28 Finiteness of 퐶푙푘………………………………………………………………....29 The Minkowski bound…………………………………………………………...32 Further remarks…………………………………………………………………..34 BIBLIOGRAPHY…………………………………………………………….………….......36 iii ACKNOWLEDGEMENTS I want to thank my advisor Dr. Gang Yu who has been very supportive, patient and encouraging throughout this tremendous and enchanting experience. Also, I want to thank my thesis committee members Dr. Ulrike Vorhauer and Dr. Stephen Gagola who help me correct mistakes I made in my thesis and provided many helpful advices. iv Chapter 1. Introduction Algebraic number theory is a branch of number theory which leads the way in the world of mathematics. It uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Concepts and results in algebraic number theory are very important in learning mathematics. -
Composition of Binary Quadratic Forms∗ Understanding the Approaches of Gauss, Dirichlet and Bhargava
GENERAL ARTICLE Composition of Binary Quadratic Forms∗ Understanding the Approaches of Gauss, Dirichlet and Bhargava François Séguin In 2004, Bhargava introduced a new way to understand the composition law of integral binary quadratic forms through what he calls the ‘cubes of integers’. The goal of this article is to introduce the reader to Bhargava’s cubes and this new composition law, as well as to relate it to the composition law as it is known classically. We will present a historical expo- sition of the subject, from Gauss to Bhargava, and see how the different formulations of the composition laws are equiv- alent. François Séguin received his PhD from Queen’s University in 2018. His research interests are in analytic number 1. Introduction theory, specifically concerning questions of Binary quadratic forms were already studied in the seventeenth algebraic nature. and eighteenth centuries. Originally, the main questions formu- lated in terms of binary quadratic forms were about the repre- sentation of numbers. Given a binary quadratic form f (x, y) = ax2 + bxy + cy2, we can ask which integers n can be written as n = f (x0, y0)forsomex0, y0, in which case f is said to represent n. Given an integer n, we can also ask which binary quadratic forms could represent n. Finally, given such a representation, how many other representations can we find. It is interesting to note that these very questions will eventually come to play a role in subjects such as Diophantine equations, quadratic reciprocity and even class field theory. Mathematicians like Fermat and Eu- ler tried to classify all the possible binary quadratic forms, but Keywords the biggest results came from Carl Friedrich Gauss in 1801 in his Cubes of integers, binary quadratic forms, composition Disquisitiones Arithmeticae. -
Counting Class Numbers
Royal Institute of Technology Stockholm University Master thesis Counting Class Numbers Tobias Magnusson supervised by P¨ar Kurlberg January 4, 2018 Acknowledgements Firstly, I would like to thank my supervisor P¨arKurlberg for suggesting such a rich problem for me to study. Secondly, I would like to thank my classmates for their support, and in particular Simon Almerstr¨om- Przybyl and Erin Small. Thirdly, I would like to thank Federico Pintore for answering important questions concerning his PhD thesis. Finally, and most importantly, I would like to thank my wife for her patience with my mathematical obsessions. 2 Abstract The following thesis contains an extensive account of the theory of class groups. First the form class group is introduced through equivalence classes of certain integral binary quadratic forms with a given discriminant. The sets of classes is then turned into a group through an operation referred to as \composition". Then the ideal class group is introduced through classes of fractional ideals in the ring of integers of quadratic fields with a given discriminant. It is then shown that for negative fundamental discriminants, the ideal class group and form class group are isomorphic. Some concrete computations are then done, after which some of the most central conjectures concerning the average behaviour of class groups with discriminant less than X { the Cohen-Lenstra heuristics { are stated and motivated. The thesis ends with a sketch of a proof by Bob Hough of a strong result related to a special case of the Cohen-Lenstra heuristics. Att r¨aknaklasstal F¨oljandemastersuppsats inneh˚alleren utf¨orlig redog¨orelseav klassgruppsteori. -
Integer Forms and Elliptic Curves
INTEGER FORMS AND ELLIPTIC CURVES YUZU IDO, IAN RUOHONIEMI, AND MATTHEW STEVENS Abstract. In this paper we develop the necessary background to un- derstand Bhargava’s 2015 result on the average rank of elliptic curves. We do this through expanding on integer forms, specifically quadratic and quartic forms, before going into depth on Galois cohomology and its role in elliptic curves. Contents 1. Introduction 1 2. When Do Forms Represent Zero? 2 2.1. Serre’s Proof of Hasse-Minkowski 3 2.2. Gauss’s Article 294 Algorithm 5 3. Invariants of Binary Forms 7 4. Binary Forms with Bounded Discriminant 8 5. An Overview of Elliptic Curves 13 6. Finite Generation of Elliptic Curve Groups 15 6.1. Mordell’s Theorem 15 6.2. Torsion 17 7. An Overview of Group Cohomology 18 7.1. Definition 18 7.2. Restriction and Galois modules 22 8. Geometric Machinery 22 8.1. Projective Curves and Their Divisors 23 8.2. Twists and Torsors 25 9. The Shafarevich-Tate and Selmer Groups 27 10. Quartic Forms and the Average Rank of Elliptic Curves 29 Acknowledgements 31 References 31 1. Introduction Few topics in algebra have as deep a historical establishment as that of integer forms. Quadratic forms, the degree two case of integer forms, have been studied since antiquity. Their theory was extensively developed by Date: August 2019. 1 2 IDO, RUOHONIEMI, AND STEVENS the work of Euler and Legendre, and then formalized and made rigorous in Gauss’ Disquisitiones. However, progress then decreased in this branch of algebra, as other topics took the forefront. -
A Friendly Introduction to Ideal Class Groups and an Interesting Problem
A Friendly Introduction to Ideal Class Groups and An Interesting Problem Sutirtha Datta,F.Y.B.Sc(CMI) July 2019 1 Introduction In this article, we will mainly deal with one of the many interesting topics of Algebraic Number Theory : Class Numbers ( Red Alert:Please don't drag CMI Alg-2 topic on conjugacy classes in a group here) and we will see a classic prob- lem known as "Gauss's class number problem for imaginary quadratic fields”, which will eventually lead us to a sweet theorem named Heegner Theorem (credit goes to Harold Stark as well) I assume the reader to know basic notions of ring,ideals,field,field extensions etc ,else they can just have a glance on my Galois Theory article.Still I will give definitions of basic notions of UFD, PID and so on below. 2 Some Preliminary Concepts 2.1 Notion of Integral Closure Before I give the definitions,I would like to ask the reader to realize the similarity between the notion of an element being algebraic "over" an algebraic extension and that of integral "over" an integral extension while going through this section. Consider R is a subring of the commutative ring S with 1 = 1S 2 R. Then S is integral over R if every s 2 S is integral over R Now what does being integral over some ring mean? Definition 2.1 (Integral Element). : An element s 2 S is integral over R if s is root of a monic polynomial in R[x]. Definition 2.2 (Integral Closure). : The integral closure of R in S is the set of elements of S that are integral over R.