Class Group Computations in Number Fields and Applications to Cryptology Alexandre Gélin
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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). -
Class Numbers of Quadratic Fields Ajit Bhand, M Ram Murty
Class Numbers of Quadratic Fields Ajit Bhand, M Ram Murty To cite this version: Ajit Bhand, M Ram Murty. Class Numbers of Quadratic Fields. Hardy-Ramanujan Journal, Hardy- Ramanujan Society, 2020, Volume 42 - Special Commemorative volume in honour of Alan Baker, pp.17 - 25. hal-02554226 HAL Id: hal-02554226 https://hal.archives-ouvertes.fr/hal-02554226 Submitted on 1 May 2020 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. Hardy-Ramanujan Journal 42 (2019), 17-25 submitted 08/07/2019, accepted 07/10/2019, revised 15/10/2019 Class Numbers of Quadratic Fields Ajit Bhand and M. Ram Murty∗ Dedicated to the memory of Alan Baker Abstract. We present a survey of some recent results regarding the class numbers of quadratic fields Keywords. class numbers, Baker's theorem, Cohen-Lenstra heuristics. 2010 Mathematics Subject Classification. Primary 11R42, 11S40, Secondary 11R29. 1. Introduction The concept of class number first occurs in Gauss's Disquisitiones Arithmeticae written in 1801. In this work, we find the beginnings of modern number theory. Here, Gauss laid the foundations of the theory of binary quadratic forms which is closely related to the theory of quadratic fields. -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry. -
Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai. -
Binary Quadratic Forms and the Ideal Class Group
Binary Quadratic Forms and the Ideal Class Group Seth Viren Neel August 6, 2012 1 Introduction We investigate the genus theory of Binary Quadratic Forms. Genus theory is a classification of all the ideals of quadratic fields k = Q(√m). Gauss showed that if we define an equivalence relation on the fractional ideals of a number field k via the principal ideals of k,anddenotethesetofequivalenceclassesbyH+(k)thatthis is a finite abelian group under multiplication of ideals called the ideal class group. The connection with knot theory is that this is analogous to the classification of the homology classes of the homology group by linking numbers. We develop the genus theory from the more classical standpoint of quadratic forms. After introducing the basic definitions and equivalence relations between binary quadratic forms, we introduce Lagrange’s theory of reduced forms, and then the classification of forms into genus. Along the way we give an alternative proof of the descent step of Fermat’s theorem on primes that are the sum of two squares, and solutions to the related problems of which primes can be written as x2 + ny2 for several values of n. Finally we connect the ideal class group with the group of equivalence classes of binary quadratic forms, under a suitable composition law. For d<0, the ideal class 1 2 BINARY QUADRATIC FORMS group of Q(√d)isisomorphictotheclassgroupofintegralbinaryquadraticforms of discriminant d. 2 Binary Quadratic Forms 2.1 Definitions and Discriminant An integral quadratic form in 2 variables, is a function f(x, y)=ax2 + bxy + cy2, where a, b, c Z.Aquadraticformissaidtobeprimitive if a, b, c are relatively ∈ prime. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
6 Ideal Norms and the Dedekind-Kummer Theorem
18.785 Number theory I Fall 2017 Lecture #6 09/25/2017 6 Ideal norms and the Dedekind-Kummer theorem In order to better understand how ideals split in Dedekind extensions we want to extend our definition of the norm map to ideals. Recall that for a ring extension B=A in which B is a free A-module of finite rank, we defined the norm map NB=A : B ! A as ×b NB=A(b) := det(B −! B); the determinant of the multiplication-by-b map with respect to an A-basis for B. If B is a free A-module we could define the norm of a B-ideal to be the A-ideal generated by the norms of its elements, but in the case we are most interested in (our \AKLB" setup) B is typically not a free A-module (even though it is finitely generated as an A-module). To get around this limitation, we introduce the notion of the module index, which we will use to define the norm of an ideal. In the special case where B is a free A-module, the norm of a B-ideal will be equal to the A-ideal generated by the norms of elements. 6.1 The module index Our strategy is to define the norm of a B-ideal as the intersection of the norms of its localizations at maximal ideals of A (note that B is an A-module, so we can view any ideal of B as an A-module). Recall that by Proposition 2.6 any A-module M in a K-vector space is equal to the intersection of its localizations at primes of A; this applies, in particular, to ideals (and fractional ideals) of A and B. -
Algebraic Number Theory Summary of Notes
Algebraic Number Theory summary of notes Robin Chapman 3 May 2000, revised 28 March 2004, corrected 4 January 2005 This is a summary of the 1999–2000 course on algebraic number the- ory. Proofs will generally be sketched rather than presented in detail. Also, examples will be very thin on the ground. I first learnt algebraic number theory from Stewart and Tall’s textbook Algebraic Number Theory (Chapman & Hall, 1979) (latest edition retitled Algebraic Number Theory and Fermat’s Last Theorem (A. K. Peters, 2002)) and these notes owe much to this book. I am indebted to Artur Costa Steiner for pointing out an error in an earlier version. 1 Algebraic numbers and integers We say that α ∈ C is an algebraic number if f(α) = 0 for some monic polynomial f ∈ Q[X]. We say that β ∈ C is an algebraic integer if g(α) = 0 for some monic polynomial g ∈ Z[X]. We let A and B denote the sets of algebraic numbers and algebraic integers respectively. Clearly B ⊆ A, Z ⊆ B and Q ⊆ A. Lemma 1.1 Let α ∈ A. Then there is β ∈ B and a nonzero m ∈ Z with α = β/m. Proof There is a monic polynomial f ∈ Q[X] with f(α) = 0. Let m be the product of the denominators of the coefficients of f. Then g = mf ∈ Z[X]. Pn j Write g = j=0 ajX . Then an = m. Now n n−1 X n−1+j j h(X) = m g(X/m) = m ajX j=0 1 is monic with integer coefficients (the only slightly problematical coefficient n −1 n−1 is that of X which equals m Am = 1). -
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,. -
Cah˙It Arf's Contribution to Algebraic Number Theory
Tr. J. of Mathematics 22 (1998) , 1 – 14. c TUB¨ ITAK˙ CAHIT˙ ARF’S CONTRIBUTION TO ALGEBRAIC NUMBER THEORY AND RELATED FIELDS by Masatoshi G. Ikeda∗ Dedicated to the memory of Professor Cahit ARF 1. Introduction Since 1939 Cahit Arf has published number of papers on various subjects in Pure Mathematics. In this note I shall try to give a brief survey of his works in Algebraic Number Theory and related fields. The papers covered in this survey are: two papers on the structure of local fields ([1], [5]), a paper on the Riemann-Roch theorem for algebraic number fields ([6]), two papers on quadratic forms ([2], [3]), and a paper on multiplicity sequences of algebraic braches ([4]). The number of the papers cited above is rather few, but the results obtained in these papers as well as his ideas and methods developed in them constitute really essential contributions to the fields. I am firmly convinced that these papers still contain many valuable suggestions and hidden possibilities for further investigations on these subjects. I believe, it is the task of Turkish mathematicians working in these fields to undertake and continue further study along the lines developed by Cahit Arf who doubtless is the most gifted mathematician Turkey ever had. In doing this survey, I did not follow the chronological order of the publications, but rather divided them into four groups according to the subjects: Structure of local fields, Riemann-Roch theorem for algebraic number fields, quadratic forms, and multiplicity sequences of algebraic braches. My aim is not to reproduce every technical detail of Cahit Arf’s works, but to make his idea in each work as clear as possible, and at the same time to locate them in the main stream of Mathematics. -
NORM GROUPS with TAME RAMIFICATION 11 Is a Homomorphism, Which Is Equivalent to the first Equation of Bimultiplicativity (The Other Follows by Symmetry)
LECTURE 3 Norm Groups with Tame Ramification Let K be a field with char(K) 6= 2. Then × × 2 K =(K ) ' fcontinuous homomorphisms Gal(K) ! Z=2Zg ' fdegree 2 étale algebras over Kg which is dual to our original statement in Claim 1.8 (this result is a baby instance of Kummer theory). Npo te that an étale algebra over K is either K × K or a quadratic extension K( d)=K; the former corresponds to the trivial coset of squares in K×=(K×)2, and the latter to the coset defined by d 2 K× . If K is local, then lcft says that Galab(K) ' Kd× canonically. Combined (as such homomorphisms certainly factor through Galab(K)), we obtain that K×=(K×)2 is finite and canonically self-dual. This is equivalent to asserting that there exists a “suÿciently nice” pairing (·; ·): K×=(K×)2 × K×=(K×)2 ! f1; −1g; that is, one which is bimultiplicative, satisfying (a; bc) = (a; b)(a; c); (ab; c) = (a; c)(b; c); and non-degenerate, satisfying the condition if (a; b) = 1 for all b; then a 2 (K×)2 : We were able to give an easy definition of this pairing, namely, (a; b) = 1 () ax2 + by2 = 1 has a solution in K: Note that it is clear from this definition that (a; b) = (b; a), but unfortunately neither bimultiplicativity nor non-degeneracy is obvious, though we will prove that they hold in this lecture in many cases. We have shown in Claim 2.11 that a less symmetric definition of the Hilbert symbol holds, namely that for all a, p (a; b) = 1 () b is a norm in K( a)=K = K[t]=(t2 − a); which if a is a square, is simply isomorphic to K × K and everything is a norm.