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. 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 points quickly. A typically mathematical book style that begins with some motivation, definitions, examples and theorems, is used throughout. I try to pause with remarks to make readers have some thoughts before moving on. The book also requires some background in undergraduate level linear algebra and elementary 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. 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. 2 Rings and Fields The abstract treatments of rings and fields using groups are presented 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. 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. 4 Modules and Noetherian Rings Modules can be considered as a generalization of vector spaces. I cover basic concepts of modules and work on free modules. Projective and injective modules are introduced. Moreover, I present the proof of the structure theorems for modules over a PID. Noetherian and Artinion rings are also explored. In the end, I demonstrate some 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. 5 Field Theory I give more details on a construction of extension fields. Also, I prepare the readers to Galois theory. Applications of Galois theory are provided in proving fundamental 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—Ajchara Harnchoowong and Yupaporn Kemprasit at Chulalongkorn University, and Ed- 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 DefinitionsandExamples . 5 1.2.2 Subgroups..................................... 7 1.2.3 Homomorphisms................................. 9 Exercises.......................................... 11 1.3 GroupActions...................................... 12 Exercises.......................................... 17 1.4 QuotientGroupsandCyclicGroups . .... 18 1.4.1 QuotientGroups ................................. 18 1.4.2 CyclicGroups................................... 20 Exercises.......................................... 23 1.5 TheSymmetricGroup .................................. 23 Exercises.......................................... 27 1.6 SylowTheorems ..................................... 28 1.6.1 Sylow p-subgroups ................................ 28 1.6.2 ApplicationsofSylowTheorems . 31 Exercises.......................................... 32 1.7 FiniteAbelianGroups ............................... ... 34 Exercises.......................................... 43 2 Rings and Fields 45 2.1 BasicConcepts .................................... .. 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,QuotientRingsandtheFieldofFractions. ........... 52 Exercises.......................................... 55 2.3 MaximalIdealsandPrimeIdeals . ..... 56 Exercises.......................................... 58 2.4 Factorizations.................................... ... 58 2.4.1 Irreducible Elements and Prime Elements . ... 58 2.4.2 UniqueFactorizationDomains. .. 59 iii Exercises.......................................... 64 2.5 PolynomialRings .................................... 66 2.5.1 PolynomialsandTheirRoots . 66 2.5.2 Factorizations in Polynomial Rings . ... 69 Exercises.......................................... 73 2.6 FieldExtensions .................................. ... 74 2.6.1 AlgebraicandTranscendentalExtensions. ...... 74 2.6.2 MoreonRootsofPolynomials. 77 Exercises.......................................... 79 3 Advanced Group Theory 81 3.1 Jordan-HölderTheorem ............................... .. 81 Exercises.......................................... 84 3.2 SolvableGroups .................................... 85 Exercises.......................................... 87 3.3 NilpotentGroups ................................... .. 87 Exercises.......................................... 91 3.4 LinearGroups...................................... 92 Exercises.......................................... 95 3.5 FreeGroupsandPresentations . ... 96 Exercises..........................................101 4 Modules and Noetherian Rings 103 4.1 Modules.......................................... 103 Exercises..........................................107 4.2 FreeModulesandMatrices ............................. 107 Exercises..........................................112 4.3 ProjectiveandInjectiveModules . .......112 Exercises..........................................118 4.4 ModulesoveraPID.................................. 119 Exercises..........................................130 4.5 NoetherianRings.................................... 130 Exercises..........................................134 4.6 ArtinianRings.....................................