Ideals and Class Groups of Number Fields
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The Class Number One Problem for Imaginary Quadratic Fields
MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM JEREMY BOOHER Gauss found 9 imaginary quadratic fields with class number one, and in the early 19th century conjectured he had found all of them. It turns out he was correct, but it took until the mid 20th century to prove this. Theorem 1. Let K be an imaginary quadratic field whose ring of integers has class number one. Then K is one of p p p p p p p p Q(i); Q( −2); Q( −3); Q( −7); Q( −11); Q( −19); Q( −43); Q( −67); Q( −163): There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner's paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner's proof was essentially correct and in fact equiv- alent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new approach based on modular curve, reducing the class number + one problem to finding special points on the modular curve Xns(n). For certain values of n, it is feasible to find all of these points. He remarks that when \N = 24 An elliptic curve is obtained. This is the level considered in effect by Heegner." Serre says nothing more, and later writers only repeat this comment. This essay will present Heegner's argument, as modernized in Cox [7], then explain Serre's strategy. -
Dimension Theory and Systems of Parameters
Dimension theory and systems of parameters Krull's principal ideal theorem Our next objective is to study dimension theory in Noetherian rings. There was initially amazement that the results that follow hold in an arbitrary Noetherian ring. Theorem (Krull's principal ideal theorem). Let R be a Noetherian ring, x 2 R, and P a minimal prime of xR. Then the height of P ≤ 1. Before giving the proof, we want to state a consequence that appears much more general. The following result is also frequently referred to as Krull's principal ideal theorem, even though no principal ideals are present. But the heart of the proof is the case n = 1, which is the principal ideal theorem. This result is sometimes called Krull's height theorem. It follows by induction from the principal ideal theorem, although the induction is not quite straightforward, and the converse also needs a result on prime avoidance. Theorem (Krull's principal ideal theorem, strong version, alias Krull's height theorem). Let R be a Noetherian ring and P a minimal prime ideal of an ideal generated by n elements. Then the height of P is at most n. Conversely, if P has height n then it is a minimal prime of an ideal generated by n elements. That is, the height of a prime P is the same as the least number of generators of an ideal I ⊆ P of which P is a minimal prime. In particular, the height of every prime ideal P is at most the number of generators of P , and is therefore finite. -
Prime Ideals in Polynomial Rings in Several Indeterminates
PROCEEDINGS OF THE AMIERICAN MIATHEMIIATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 67 -74 S 0002-9939(97)03663-0 PRIME IDEALS IN POLYNOMIAL RINGS IN SEVERAL INDETERMINATES MIGUEL FERRERO (Communicated by Ken Goodearl) ABsTrRAcr. If P is a prime ideal of a polynomial ring K[xJ, where K is a field, then P is determined by an irreducible polynomial in K[x]. The purpose of this paper is to show that any prime ideal of a polynomial ring in n-indeterminates over a not necessarily commutative ring R is determined by its intersection with R plus n polynomials. INTRODUCTION Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials. -
Introduction to Class Field Theory and Primes of the Form X + Ny
Introduction to Class Field Theory and Primes of the Form x2 + ny2 Che Li October 3, 2018 Abstract This paper introduces the basic theorems of class field theory, based on an exposition of some fundamental ideas in algebraic number theory (prime decomposition of ideals, ramification theory, Hilbert class field, and generalized ideal class group), to answer the question of which primes can be expressed in the form x2 + ny2 for integers x and y, for a given n. Contents 1 Number Fields1 1.1 Prime Decomposition of Ideals..........................1 1.2 Basic Ramification Theory.............................3 2 Quadratic Fields6 3 Class Field Theory7 3.1 Hilbert Class Field.................................7 3.2 p = x2 + ny2 for infinitely n’s (1)........................8 3.3 Example: p = x2 + 5y2 .............................. 11 3.4 Orders in Imaginary Quadratic Fields...................... 13 3.5 Theorems of Class Field Theory.......................... 16 3.6 p = x2 + ny2 for infinitely many n’s (2)..................... 18 3.7 Example: p = x2 + 27y2 .............................. 20 1 Number Fields 1.1 Prime Decomposition of Ideals We will review some basic facts from algebraic number theory, including Dedekind Domain, unique factorization of ideals, and ramification theory. To begin, we define an algebraic number field (or, simply, a number field) to be a finite field extension K of Q. The set of algebraic integers in K form a ring OK , which we call the ring of integers, i.e., OK is the set of all α 2 K which are roots of a monic integer polynomial. In general, OK is not a UFD but a Dedekind domain. -
Faithful Abelian Groups of Infinite Rank Ulrich Albrecht
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 1, May 1988 FAITHFUL ABELIAN GROUPS OF INFINITE RANK ULRICH ALBRECHT (Communicated by Bhama Srinivasan) ABSTRACT. Let B be a subgroup of an abelian group G such that G/B is isomorphic to a direct sum of copies of an abelian group A. For B to be a direct summand of G, it is necessary that G be generated by B and all homomorphic images of A in G. However, if the functor Hom(A, —) preserves direct sums of copies of A, then this condition is sufficient too if and only if M ®e(A) A is nonzero for all nonzero right ¿ï(A)-modules M. Several examples and related results are given. 1. Introduction. There are only very few criteria for the splitting of exact sequences of torsion-free abelian groups. The most widely used of these was given by Baer in 1937 [F, Proposition 86.5]: If G is a pure subgroup of a torsion-free abelian group G, such that G/C is homogeneous completely decomposable of type f, and all elements of G\C are of type r, then G is a direct summand of G. Because of its numerous applications, many attempts have been made to extend the last result to situations in which G/C is not completely decomposable. Arnold and Lady succeeded in 1975 in the case that G is torsion-free of finite rank. Before we can state their result, we introduce some additional notation: Suppose that A and G are abelian groups. -
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). -
MAXIMAL and NON-MAXIMAL ORDERS 1. Introduction Let K Be A
MAXIMAL AND NON-MAXIMAL ORDERS LUKE WOLCOTT Abstract. In this paper we compare and contrast various prop- erties of maximal and non-maximal orders in the ring of integers of a number field. 1. Introduction Let K be a number field, and let OK be its ring of integers. An order in K is a subring R ⊆ OK such that OK /R is finite as a quotient of abelian groups. From this definition, it’s clear that there is a unique maximal order, namely OK . There are many properties that all orders in K share, but there are also differences. The main difference between a non-maximal order and the maximal order is that OK is integrally closed, but every non-maximal order is not. The result is that many of the nice properties of Dedekind domains do not hold in arbitrary orders. First we describe properties common to both maximal and non-maximal orders. In doing so, some useful results and constructions given in class for OK are generalized to arbitrary orders. Then we de- scribe a few important differences between maximal and non-maximal orders, and give proofs or counterexamples. Several proofs covered in lecture or the text will not be reproduced here. Throughout, all rings are commutative with unity. 2. Common Properties Proposition 2.1. The ring of integers OK of a number field K is a Noetherian ring, a finitely generated Z-algebra, and a free abelian group of rank n = [K : Q]. Proof: In class we showed that OK is a free abelian group of rank n = [K : Q], and is a ring. -
Algebraic Number Theory
Algebraic Number Theory Sergey Shpectorov January{March, 2010 This course in on algebraic number theory. This means studying problems from number theory with methods from abstract algebra. For a long time the main motivation behind the development of algebraic number theory was the Fermat Last Theorem. Proven in 1995 by Wiles with the help of Taylor, this theorem states that there are no positive integers x, y and z satisfying the equation xn + yn = zn; where n ≥ 3 is an integer. The proof of this statement for the particular case n = 4 goes back to Fibonacci, who lived four hundred years before Fermat. Modulo Fibonacci's result, Fermat Last Theorem needs to be proven only for the cases where n = p is an odd prime. By the end of the course we will hopefully see, as an application of our theory, how to prove the Fermat Last Theorem for the so-called regular primes. The idea of this belongs to Kummer, although we will, of course, use more modern notation and methods. Another accepted definition of algebraic number theory is that it studies the so-called number fields, which are the finite extensions of the field of ra- tional numbers Q. We mention right away, however, that most of this theory applies also in the second important case, known as the case of function fields. For example, finite extensions of the field of complex rational functions C(x) are function fields. We will stress the similarities and differences between the two types of fields, as appropriate. Finite extensions of Q are algebraic, and this ties algebraic number the- ory with Galois theory, which is an important prerequisite for us. -
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.