Commutative Rings and 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. -
Algebra I (Math 200)
Algebra I (Math 200) UCSC, Fall 2009 Robert Boltje Contents 1 Semigroups and Monoids 1 2 Groups 4 3 Normal Subgroups and Factor Groups 11 4 Normal and Subnormal Series 17 5 Group Actions 22 6 Symmetric and Alternating Groups 29 7 Direct and Semidirect Products 33 8 Free Groups and Presentations 35 9 Rings, Basic Definitions and Properties 40 10 Homomorphisms, Ideals and Factor Rings 45 11 Divisibility in Integral Domains 55 12 Unique Factorization Domains (UFD), Principal Ideal Do- mains (PID) and Euclidean Domains 60 13 Localization 65 14 Polynomial Rings 69 Chapter I: Groups 1 Semigroups and Monoids 1.1 Definition Let S be a set. (a) A binary operation on S is a map b : S × S ! S. Usually, b(x; y) is abbreviated by xy, x · y, x ∗ y, x • y, x ◦ y, x + y, etc. (b) Let (x; y) 7! x ∗ y be a binary operation on S. (i) ∗ is called associative, if (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x; y; z 2 S. (ii) ∗ is called commutative, if x ∗ y = y ∗ x for all x; y 2 S. (iii) An element e 2 S is called a left (resp. right) identity, if e ∗ x = x (resp. x ∗ e = x) for all x 2 S. It is called an identity element if it is a left and right identity. (c) S together with a binary operation ∗ is called a semigroup, if ∗ is as- sociative. A semigroup (S; ∗) is called a monoid if it has an identity element. 1.2 Examples (a) Addition (resp. -
Exercises and Solutions in Groups Rings and Fields
EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuo˘glu Middle East Technical University [email protected] Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . v 1. SETS, INTEGERS, FUNCTIONS . 1 2. GROUPS . 4 3. RINGS . .55 4. FIELDS . 77 5. INDEX . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract al- gebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank B¨ulent B¨uy¨ukbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. Ismail_ S¸. G¨ulo˘glufor checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and G¨uldaneG¨um¨u¸sfor their typing the manuscript in LATEX. Mahmut Kuzucuo˘glu I would like to thank our graduate students Tu˘gbaAslan, B¨u¸sra C¸ınar, Fuat Erdem and Irfan_ Kadık¨oyl¨ufor reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuo˘glu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2n distinct subsets. -
Boolean and Abstract Algebra Winter 2019
Queen's University School of Computing CISC 203: Discrete Mathematics for Computing II Lecture 7: Boolean and Abstract Algebra Winter 2019 1 Boolean Algebras Recall from your study of set theory in CISC 102 that a set is a collection of items that are related in some way by a common property or rule. There are a number of operations that can be applied to sets, like [, \, and C. Combining these operations in a certain way allows us to develop a number of identities or laws relating to sets, and this is known as the algebra of sets. In a classical logic course, the first thing you typically learn about is propositional calculus, which is the branch of logic that studies propositions and connectives between propositions. For instance, \all men are mortal" and \Socrates is a man" are propositions, and using propositional calculus, we may conclude that \Socrates is mortal". In a sense, propositional calculus is very closely related to set theory, in that propo- sitional calculus is the study of the set of propositions together with connective operations on propositions. Moreover, we can use combinations of connective operations to develop the laws of propositional calculus as well as a collection of rules of inference, which gives us even more power to manipulate propositions. Before we continue, it is worth noting that the operations mentioned previously|and indeed, most of the operations we have been using throughout these notes|have a special name. Operations like [ and \ apply to pairs of sets in the same way that + and × apply to pairs of numbers. -
On Free Quasigroups and Quasigroup Representations Stefanie Grace Wang Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 On free quasigroups and quasigroup representations Stefanie Grace Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Wang, Stefanie Grace, "On free quasigroups and quasigroup representations" (2017). Graduate Theses and Dissertations. 16298. https://lib.dr.iastate.edu/etd/16298 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On free quasigroups and quasigroup representations by Stefanie Grace Wang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Jonathan D.H. Smith, Major Professor Jonas Hartwig Justin Peters Yiu Tung Poon Paul Sacks The student author and the program of study committee are solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2017 Copyright c Stefanie Grace Wang, 2017. All rights reserved. ii DEDICATION I would like to dedicate this dissertation to the Integral Liberal Arts Program. The Program changed my life, and I am forever grateful. It is as Aristotle said, \All men by nature desire to know." And Montaigne was certainly correct as well when he said, \There is a plague on Man: his opinion that he knows something." iii TABLE OF CONTENTS LIST OF TABLES . -
7.2 Binary Operators Closure
last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers. -
On Field Γ-Semiring and Complemented Γ-Semiring with Identity
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 8(2018), 189-202 DOI: 10.7251/BIMVI1801189RA Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) ON FIELD Γ-SEMIRING AND COMPLEMENTED Γ-SEMIRING WITH IDENTITY Marapureddy Murali Krishna Rao Abstract. In this paper we study the properties of structures of the semi- group (M; +) and the Γ−semigroup M of field Γ−semiring M, totally ordered Γ−semiring M and totally ordered field Γ−semiring M satisfying the identity a + aαb = a for all a; b 2 M; α 2 Γand we also introduce the notion of com- plemented Γ−semiring and totally ordered complemented Γ−semiring. We prove that, if semigroup (M; +) is positively ordered of totally ordered field Γ−semiring satisfying the identity a + aαb = a for all a; b 2 M; α 2 Γ, then Γ-semigroup M is positively ordered and study their properties. 1. Introduction In 1995, Murali Krishna Rao [5, 6, 7] introduced the notion of a Γ-semiring as a generalization of Γ-ring, ring, ternary semiring and semiring. The set of all negative integers Z− is not a semiring with respect to usual addition and multiplication but Z− forms a Γ-semiring where Γ = Z: Historically semirings first appear implicitly in Dedekind and later in Macaulay, Neither and Lorenzen in connection with the study of a ring. However semirings first appear explicitly in Vandiver, also in connection with the axiomatization of Arithmetic of natural numbers. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
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. -
Chapter 4 Complex Numbers Course Number
Chapter 4 Complex Numbers Course Number Section 4.1 Complex Numbers Instructor Objective: In this lesson you learned how to perform operations with Date complex numbers. Important Vocabulary Define each term or concept. Complex numbers The set of numbers obtained by adding real number to real multiples of the imaginary unit i. Complex conjugates A pair of complex numbers of the form a + bi and a – bi. I. The Imaginary Unit i (Page 328) What you should learn How to use the imaginary Mathematicians created an expanded system of numbers using unit i to write complex the imaginary unit i, defined as i = Ö - 1 , because . numbers there is no real number x that can be squared to produce - 1. By definition, i2 = - 1 . For the complex number a + bi, if b = 0, the number a + bi = a is a(n) real number . If b ¹ 0, the number a + bi is a(n) imaginary number .If a = 0, the number a + bi = bi is a(n) pure imaginary number . The set of complex numbers consists of the set of real numbers and the set of imaginary numbers . Two complex numbers a + bi and c + di, written in standard form, are equal to each other if . and only if a = c and b = d. II. Operations with Complex Numbers (Pages 329-330) What you should learn How to add, subtract, and To add two complex numbers, . add the real parts and the multiply complex imaginary parts of the numbers separately. numbers Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. -
Factorization Theory in Commutative Monoids 11
FACTORIZATION THEORY IN COMMUTATIVE MONOIDS ALFRED GEROLDINGER AND QINGHAI ZHONG Abstract. This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold. 1. Introduction Factorization theory emerged from algebraic number theory. The ring of integers of an algebraic number field is factorial if and only if it has class number one, and the class group was always considered as a measure for the non-uniqueness of factorizations. Factorization theory has turned this idea into concrete results. In 1960 Carlitz proved (and this is a starting point of the area) that the ring of integers is half-factorial (i.e., all sets of lengths are singletons) if and only if the class number is at most two. In the 1960s Narkiewicz started a systematic study of counting functions associated with arithmetical properties in rings of integers. Starting in the late 1980s, theoretical properties of factorizations were studied in commutative semigroups and in commutative integral domains, with a focus on Noetherian and Krull domains (see [40, 45, 62]; [3] is the first in a series of papers by Anderson, Anderson, Zafrullah, and [1] is a conference volume from the 1990s). -
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.