Abstract Algebra Groups, Rings and Fields, Advanced Group Theory, Modules and Noetherian Rings, Field Theory YOTSANAN MEEMARK Semi-formal based on the graduate courses 2301613–4 Abstract Algebra I & II, offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University Published by Yotsanan Meemark Department of Mathematics and Computer Science Faculty of Science, Chulalongkorn University, Bangkok, 10330 Thailand First digital edition October 2013 First bound edition May 2014 Second bound edition August 2015 Available for free download at http://pioneer.netserv.chula.ac.th/~myotsana/ Please cite this book as: Y. Meemark, Abstract Algebra, 2015, PDF available at http://pioneer.netserv.chula.ac.th/~myotsana/ Any comment or suggestion, please write to [email protected] c 2015 by Yotsanan Meemark. (cid:13) Meemark, Yotsanan Abstract Algebra / Yotsanan Meemark – 2nd ed. Bangkok: Danex Intercorporation Co., Ltd., 2015. 195pp. ISBN 978-616-361-389-9 Printed by Danex Intercorporation Co., Ltd., Bangkok, Thailand. www.protexts.com Foreword This book is written based on two graduate abstract algebra courses offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University. It grows out of my lecture notes that I used while I was teaching those courses many times. My intention is to develop essential topics in algebra that can be used in research as illustrated some in the final chapter. Also, it can be served as a standard reference for preparing for a qualifying examination in Algebra. I have tried to make it self-contained as much as possible. However, it may not be suitable for reading it for the first course in abstract algebra. It hits and goes through many basic pointsquickly. Atypicallymathematicalbookstylethatbeginswithsomemotivation,definitions, examples and theorems, is used throughout. I try to pause with remarks to make readers have some thoughts before moving on. Thebookalsorequiressomebackgroundinundergraduatelevellinearalgebraandelementary number theory. For example, I assume the readers to have known matrix theory over a field in which treatment can be found in most linear algebra books. My number theory lecture note is available on the web-page as well. However, some essential results are recalled in the first section. I give many examples to demonstrate new definitions and theorems. In addition, when the converse of a theorem may not hold, counter examples are provided. The major points are divided into five chapters as follows. (cid:4) 1 Groups A group is a basic algebraic structure but it is a core in this course. I choose the approach via group actions. Although it is not quite elementary, it is an important aspect in dealing with groups. I also cover Sylow theorems with some applications on finite groups. The structure theorem of finite abelian groups is also presented. (cid:4) 2RingsandFields Theabstracttreatmentsofringsandfieldsusinggroupsarepresented in the first section. Rings discussed throughout this book always contain the identity. Ideals and factorizations are discussed in detail. In addition, I talk about polynomials over a ring and which will be used in a construction of field extensions. (cid:4) 3 Advanced Group Theory In this chapter, I give deeper theory of groups. Various kinds of series of a group are studied in the first three sections. I also have results on a linear group. Finally, I show how to construct a group from a set of objects and presentations and talk about a graphical representation called a Cayley graph. (cid:4) 4ModulesandNoetherianRings Modulescanbeconsideredasageneralizationofvector spaces. Icoverbasicconceptsofmodulesandworkonfreemodules. Projectiveandinjective modulesareintroduced. Moreover,Ipresenttheproofofthestructuretheoremsformodules overaPID.NoetherianandArtinionringsarealsoexplored. Intheend,Idemonstratesome aspects in doing research in algebra. The readers will see some applications of module theory, especially a free R-module over commutative rings, to obtain a structure theorem for finite dimensional symplectic spaces over a local ring. The symplectic graphs over a commutative ring is defined and studied. (cid:4) 5FieldTheory Igivemoredetailsonaconstructionofextensionfields. Also,Ipreparethe readerstoGaloistheory. ApplicationsofGaloistheoryareprovidedinprovingfundamental theorem of algebra, finite fields, and cyclotomic fields. For the sake of completeness, I discuss some results on a transcendental extension in the final section. i The whole book is designed for a year course. Chapters 1 and 2 are appropriate for a first course and Chapters 3, 4 and 5 can be served as a more advanced course. There are many topics that, in my opinion, they are worth mentioned. I try to break each topic in step-by-step and scatter it as a “Project” throughout this book. The projects consist of lengthy/generalization exercises, computations of numerical examples, programming sugges- tions, and research/open problems. This allows us to see that abstract algebra has many applica- tions and is still an active subject. They are independent and can be skipped without any effects on the continuity of the reading. The book would not have been possible without great lectures from my abstract algebra teachers—AjcharaHarnchoowongandYupapornKemprasitatChulalongkornUniversity,andEd- ward Formanek at the Pennsylvania State University. They initiate wonderful resources to com- pose each section in this book. I express my gratitude to them all. I take full responsibility for typos/mistakes that may be found in the manuscript. If you catch ones or have any other suggestions, please write to me. I shall include and correct them in the more up-to-date version once a year on the website. Finally, I hope that the textbook will benefit many students, teachers and researchers in Algebra and Number Theory. Yotsanan Meemark Bangkok, Thailand ii Contents Foreword i Contents iii 1 Groups 1 1.1 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Quotient Groups and Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.1 Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.2 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 The Symmetric Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.6 Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6.1 Sylow p-subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6.2 Applications of Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . 31 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 Rings and Fields 45 2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.3 Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.1.4 Ring Homomorphisms and Group Rings . . . . . . . . . . . . . . . . . . . . 50 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2 Ideals, Quotient Rings and the Field of Fractions. . . . . . . . . . . . . . . . . . . . 52 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3 Maximal Ideals and Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4 Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.1 Irreducible Elements and Prime Elements . . . . . . . . . . . . . . . . . . . 58 2.4.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 59 iii Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.5 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.5.1 Polynomials and Their Roots . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.5.2 Factorizations in Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . 69 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.6 Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.6.1 Algebraic and Transcendental Extensions . . . . . . . . . . . . . . . . . . . . 74 2.6.2 More on Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3 Advanced Group Theory 81 3.1 Jordan-Hölder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.3 Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4 Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.5 Free Groups and Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4 Modules and Noetherian Rings 103 4.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Free Modules and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3 Projective and Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.4 Modules over a PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.5 Noetherian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6 Artinian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.7 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.7.1 Symplectic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.7.2 Symplectic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5 Field Theory 145 5.1 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.2 Algebraic Closure of a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3 Multiple Roots and Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.4 Automorphisms of Fields and Galois Theory . . . . . . . . . . . . . . . . . . . . . . 155 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.5 Some Consequences of Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.6 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 iv 5.7 Cyclotomic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.8 Normal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.9 Transcendental Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Bibliography 183 Index 185 v This page intentionally left blank 1 | Groups We write N for the set of positive integers, Z for the set of integers, Q for the set of rational numbers, R for the set of real numbers and C for the set of complex numbers. In this first chapter, we talk about a group which is a basic algebraic structure. However, it is a core in this course. Our approach here relies on group actions. Although it is not quite elementary, it is an important aspect in dealing with groups. We also discuss Sylow theorems with some applications and the structure of finite abelian groups. 1.1 Integers As a number theorist, before I jump into the abstract part, let’s lay down some foundations. My first undergraduate abstract algebra course started with elementary number theory—the study of integers. It contains many examples to bear in mind while we are studying the more general results in other abstract domains. Theorem 1.1.1. [Division Algorithm] Given integers a and b, with b = 0, there exist unique 6 integers q and r satisfying a = qb+r, where 0 r < b . ≤ | | Theintegersq andr arecalled,respectively,thequotientandremainderinthedivisionofabyb. Proof. To prove this theorem, we must use the well-ordering principle, namely, “every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a in S such that a b for all b S”. ≤ ∈ Existence: Firstweshallassumethatb > 0. LetS = a xb : x Z and a xb 0 N 0 . { − ∈ − ≥ } ⊆ ∪{ } We shall show that S = . Since b 1, we have a b a , so 6 ∅ ≥ | | ≥ | | a ( a )b = a+ a b a+ a 0, − −| | | | ≥ | | ≥ Then a ( a )b S, so S = . By the well-ordering principle, S contains a least element, call − −| | ∈ 6 ∅ it r. Then a qb = r for some q Z. Since r S, r 0 and a = qb+r. It remains to show that − ∈ ∈ ≥ r < b. Suppose that r b. Thus, ≥ 0 r b = a qb b = a (q+1)b, ≤ − − − − so r b r and r b S. This contradicts the minimality of r. Hence, r < b. − ≤ − ∈ Next, we consider the case in which b < 0. Then b > 0 and Theorem 2.5.2 gives q′,r Z | | ∈ such that a = q′ b +r, where 0 r < b . | | ≤ | | Since b = b, we may take q = q′ to arrive at | | − − a = qb+r, where 0 r < b ≤ | | 1 2 1. Groups as desired. Uniqueness: Let q,q′,r,r′ Z be such that ∈ a = qb+r and a = q′b+r′, where 0 r,r′ < b . Then ≤ | | (q q′)b = r′ r. − − Since 0 r,r′ < b , we have r′ r < b , so b q q′ = r′ r < b . This implies that ≤ | | | − | | | | || − | | − | | | 0 q q′ < 1, hence q = q′ which also forces r = r′. ≤ | − | Theorem1.1.1providesanimportantexampleinSection2.4wherewediscussamoregeneral domain called a Euclidean domain. An integer b is said to be divisible by an integer a = 0, in symbols a b, if there exists some 6 | integer c such that b = ac. We write a ∤ b to indicate that b is not divisible by a. Note that a b a b, so we may consider only positive divisors. The next theorem contains elementary | ⇔ − | properties of divisibility. Theorem 1.1.2. For integers a,b and c, the following statements hold: 1. a 0, 1 a, a a. | | | 2. a 1 if and only if a = 1. | ± 3. If a b, then a ( b), ( a) b and ( a) ( b). | | − − | − | − 4. If a b and c d, then ac bd. | | | 5. If a b and b c, then a c. | | | 6. (a b and b a) if and only if a = b. | | ± 7. If a b and b = 0, then a b . | 6 | | ≤ | | 8. If a b and a c, then a (bx+cy) for arbitrary integers x and y. | | | An integer p > 1 is called a prime number, or simply a prime, if its only positive divisors are 1 and p. An integer greater than 1 which is not a prime is termed composite. Example 1.1.1. 2,3,5,11,2011 are primes. 6,8,12,2558 are composite numbers. Let a and b be given integers, with at least one of them different from zero. The greatest common divisor (gcd) of a and b, denoted by gcd(a,b), is the positive integer d satisfying: 1. d a and d b, 2. for all c Z, if c a and c b, then c d. | | ∈ | | ≤ Basic properties of gcd are collected in the next theorem. Theorem 1.1.3. Let a and n be integers not both zero. 1. If d = min ax+ny > 0 : x,y Z , then d = gcd(a,n). { ∈ } 2. If gcd(a,n) = d, then x,y Z,ax+ny = d. ∃ ∈ 3. gcd(a,n) = 1 x,y Z,ax+ny = 1. ⇔ ∃ ∈ Proof. (1) The given set contains a2 + n2, so it is not empty and d exists by the well-ordering principle. Then d = ax+ny > 0 for some x,y Z. We shall prove that d = gcd(a,n). By the ∈ division algorithm, q,r Z,a = dq+r with 0 r d. If r > 0, then ∃ ∈ ≤ ≤ 0 < r = a dq = a (ax+ny)q = a(1 xq) nyq < d − − − − whichcontradictstheminimalityofd. Hence,r = 0andd n. Similarly,d n. Sinced = ax+ny, | | gcd(a,n) d, so gcd(a,n) d. But d a and d n, so d gcd(a,n). Hence, d = gcd(a,n). (2) | ≤ | | ≤ follows from (1) and (3) follows from (2). The converse of (3) is immediate.