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ABSTRACT ALGEBRA An Introduction ABSTRACT ALGEBRA An Introduction THOMAS W. HUNGERFORD Cleveland State University É 4 SAUNDERS COLLEGE PUBLISHING Philadelphia Ft. Worth Chicago San Francisco Montreal Toronto London Sydney Tokyo Preface one another. The congruence theme is strongly emphasized in the develop- ment of quotient rings. ConsequentIy, students can see that ideals, cosets, and quotient rings are a natural extension of familiar concepts in the in- tegers rather than an unmotivated mystery. Dealing with congruence in polynomial domains before arbitrary rings not only eases the transition but also provides some nontrivial results on field extensions much earlier than customary. To assist the student in forming a coherent overview, the interconnec- tions of the basic areas of algebra are frequently pointed out in the text and in the THEMATIC TABLE OF CONTENTS on page xx. Each horizontal line of that table shows how a particular theme or subtheme is developed first for integers, then for polynomials, rings, and groups. Each vertical column exhibits the interplay of all the themes for a particular topic. The final version of this book has benefited from the comments of many students and of mathematicians who reviewed or class-tested the manuscript. My thanks to: Gary D. Crown, Wichita State University Richard Grassl, University of New Mexico Paul R. Halmos, Santa Clara University Robin Hartshorne, University of California, Berkeley Peter Jones, Marquette University Robert Lax, Louisiana State University David Leep, University of Kentucky Arthur Lieberman, Cleveland State University Steve Monk, University of Washington Philip Montgomery, University of Kansas Murray Schacher, University of California, Los Angeles Daniel B. Shapiro, Ohio State University Nick Vaughan, University of North Texas Bhushan Wadhwa, Cleveland State University Ialso want to thank Ann Melville and Joyce Pluth for expertly typing the first draft and patiently assisting me to learn word processing for later drafts. The latter would not have been possible without the computer pro- vided by Dean Georgia Lesh-Laurie and extensive technical assistance from my colleague Frank Lozier. Their generosity is appreciated. I am also happy to acknowledge the assistance of Bob Stern, Maureen Tannuzzi, and the rest of the Saunders College Publishing staff. It has been a pleasure to work with them. Finally, a very special thank you goes to my wife, Mary Alice, and my children, Anne and Tom, for their patience and understanding during the preparation of this book. T.w.H. TOPICAL TABLE OF CONTENTS TO THE INSTRUCTOR xiii TO THE STUDENT xvii THEMATIC TABLE OF CONTENTS xx PART 1 THE CORE COURSE 1 ARITHMETIC IN Z REVISITED 1 1.1. The Division Algorithm 1 1.2. Divisibility 6 1.3. Primes and Unique Factorization 13 1.4.* Primality Testing 19 2 CONGRUENCE IN Z AND MODULAR ARITHMETIC 23 2.1. Congruence and Congruence Classes 23 2.2. Modular Arithmetic 29 2.3. The Structure of Z, When pls Prime 36 3 RINGS 39 31. Definition and Examples of Rings 40 3.2. Basic Properties of Rings 53 3.3. Isomorphism 62 * Sections and chapters in The Core Course marked * may be omitted or postponed. See the beginning of each such section for specifies. topical table ot Contents PART 2 ADVANCED TOPICS 9 FIELD EXTENSIONS 9.1. Bases and Dimension 268 9.2. Simple Extensions 276 9.3. —Algebraic Extensions 283 9.4. Splitting Fields 288 9.5. Separability 295 9.6. Finite Fields 299 10 TOPICS IN GROUP THEORY 10.1. Direct Products 308 10.2. Finite Abelian Groups 317 10.3. The Sylow Theorems and Applications 326 10.4. Conjugacy and the Proof of the Sylow Theorems 332 10.5. The Simplicity of A, 340 11 GALOIS THEORY 11.1. The Galois Group 346 11.2. The Fundamental Theorem of Galois Theory 353 11.3. Solvability by Radicals 361 PART 3 EXCURSIONS AND APPLICATIONS 12 PUBLIC-KEY CRYPTOGRAPHY Prerequisite: Section 2.3 13 THE CHINESE REMAINDER THEOREM 13.1. Proof of the Chinese Remainder Theorem 379 Prerequisites: Section 2.1, Appendix C 13.2. Applications of the Chinese Remainder Theorem 385 Prerequisite: Section 3.1 13.3. The Chinese Remainder Theorem for Rings 389 Prerequisite: Chapter 6 14 LATTICES AND BOOLEAN ALGEBRAS Prerequisites: Chapter 3, Appendices A and B 268 308 345 373 379 394 xi Topical Table of Contents 14.1. lattices 394 14.2. Boolean Algebras 404 14.3. Applications of Boolean Algebras 416 15 GEOMETRIC CONSTRUCTIONS 425 Prerequisites: Sections 4.1, 4.4, 4.5 16 ALGEBRAIC CODING THEORY 437 16.1. Linear Codes 438 Prerequisites: Section 7.4, Appendix F 16.2. Decoding Techniques 449 Prerequisite: Section 7.8 (through Lagrange's Theorem) 16.3. BCH Codes 459 Prerequisite: Section 9.6 APPENDICES A. LOGIC AND PROOF 469 B. SETS AND FUNCTIONS 481 €. WELL ORDERING AND INDUCTION 497 D. EQUIVALENCE RELATIONS 507 E. THE BINOMIAL THEOREM 513 F. MATRIX ALGEBRA 517 G. POLYNOMIALS 523 BIBLIOGRAPHY 5H ANSWERS AND SUGGESTIONS FOR SELECTED ODD-NUMBERED EXERCISES 535 INDEX 565 xviii To The Student logic and reading or writing proofs, you should begin by reading Appendix A, It summarizes the basic rules of logic and the proof techniques that are used throughout this book, Read the text with pencil and paper in hand before looking at the exercíses. When you read the statement of a theorem, be sure you know Le meaning of all the terms in the statement of the theorem. For example, ifit says “every finite integral domain is a field,” review the definitions of “integral domain” and “field” — if necessary, use the index to find the definitions. Once you understand what the theorem claims is true, then turn to the proof. Remember: There is a great deal of difference between understand- inga proofin the text and constructing one yourself. Just as you can appreci- ate a new building without being an architect, you can verify the validity of proofs presented by others even if you can't see how anyone ever thought of doing it this way in the first place. Begin by skimming through the proof to get an idea of its general outline before worrying about the details in each step. It's easier to under- stand an argument if you know approximately where it's headed. Then go back to the beginning and read the proof carefully, line by line, If it says “such and such is true by Theorem 5.18,” check to see just what Theorem 5.18 says and be sure you understand why it applies here. Ifyou get stuck, take that part on faith and finish the rest of the proof, Then go back and seeif you can figure out the sticky point, When you're really stuck, ask your instructor. He or she will welcome questions that arise from a serious effort on your part. EXERCISES Mathematics is not a spectator sport. You can't expect to learn mathematics without doing mathematics, any more than you could learn to swim without getting in the water. That's why there are so many exercisesin this book. The exercises in group A are usually straightforward. If you can't do almost all of them, you don't really understand the material. The exercises in group B often require a reasonable amount of thought — and for most of us, some trial and error as well. But the vast majority ofthem are within your grasp. The exercises in group € are usually difficult . . . a good test for strong students. Many of the exercises will ask you to prove something. As you build up your skillin reading the proofs of others (as discussed above), you will find it easier to make proofs of your own. The proof techniques presented in Appendix A may also be helpful, Answers (or hints) for approximately half of the odd-numbered exer- cises are given at the end of the book. (140) werosd, sapon sôung soy MO 20,j0 Jepurewoy Hod HO 241 suupronddy esouno €9T ger set Te I 1 1 l l árooN suojsuayxa | | spo Pita t 6 I I | ] l Í sueuog stumg teaoquy ur uegon?) big ur blzu Zu zu nounpuy | e) puesjeopy | e oouansíuoo | e) anamguy | e) sum | e) conóniduos | el onstiguy 8 9 s F Ka z 1 T l ároSyL | dnorsy wsadoy | ot | + y fiosyp seagogty fuipoy suorpnujsuos ueojog Aydeitodao oreigoity >ujamoss pue seonye Áaypogqua EL TO st “PI g1 xv xviii To The Student logic and reading or writing proofs, you should begin by reading Appendix A. It summarizes the basic rules of logic and the proof techniques that are used throughout this book. Read the text with pencil and paper in hand before looking at the exercises. When you read the statement ofa theorem, be sure you know the meaning of all the terms in the statement of the theorem. For example, ifit says “every finite integral domain is a field,” review the definitions of “integral domain” and “field” — if necessary, use the index to find the definitions. Once you understand what the theorem claims is true, then turnto the proof. Remember: There is a great deal of difference between understand- ing a proofin the text and constructing one yourself. Just as you can appreci- ate anew building without being an architect, you can verify the validity of proofs presented by others even ifyou can't see how anyone ever thought of doing it this way in the first place. Begin by skimming through the proof to get an idea of its general outline before worrying about the details in each step. It's easier to under- stand an argument if you know approximately where it's headed. Then go back to the beginning and read the proof carefully, line by line. If it says “such and such is true by Theorem 5.18,” check to see just what Theorem 5.18 says and be sure you understand why it applies here. If you get stuck, take that part on faith and finish the rest ofthe proof. Then go back and seeif you can figure out the sticky point. When you're really stuck, ask your instructor. He or she will welcome questions that arise from a serious effort on your part. EXERCISES Mathematics is not a spectator sport. You can't expect to learn mathematics without doing mathematics, any more than you could learn to swim without getting in the water. That's why there are so many exercisesin this book. The exercises in group A are usually straightforward. If you can't do almost all of them, you don't really understand the material. The exercises in group B often require a reasonable amount of thought — and for most of us, some trial anderror as well. But the vast majority ofthem are within your grasp. The exercises in group C are usually difficult . . . a good test for strong students. Many ofthe excrcises willask you to prove something. As you build up your skiltin reading the proofs of others (as discussed above), you will find it easier to make proofs of your own. The proof techniques presented in Appendix A may also be helpful. Answers (or hints) for approximately half of the odd-numbered exer- cises are given at the end of the book. To fhe Student KEEPING IT ALL STRAIGHT The different branches of algebra are con- nected in a myriad of ways that are not always clear to either the beginner or the experienced practitioner. So it's no wonder that students sometimes have trouble seeing how the various topícs tie together, or even ifthey do. Keeping in mind the three themes mentioned above will help, especially if you regularly consult the THEMATIC TABLE OF CONTENTS onthe next page. Unlike the usual table of contents, which lists the topics in numerical order by chapter and section, the Thematie Table is arranged in logical order, so you can see how everything fits together. xix DIRECTIONS: Generally speaking, ach block represents a generalization of the block immediately to its left. Reading from left to right shows how the theme or subtheme listed in the left-hand column is developed first in the integers, then in polynomials, and finally in rings and groups. Each vertical columm shows how all the themes are carried out for the topic listed at the top of the column. RINGS GROUPS 8. Arithmetic in integra! Domains .2, (Last Part) Euclidean Domains 8.1, Unique Factorization Domains 8.2. (First Past) Unique Factorization and Principal Ideals 8.3. Factorization of Quadratic Integers 6. Ideals and Quotient Rings 7. Groups 6.1. tdeals and Congruence 7.5. Congruence and Normal Subgroups 6.2. Quotient Rings and Homomorphisms 7.6. Quotient Groups 77. Quotien Groups and Homomorphisms 6.3. The Structure of R// When /1s Prime or Maximal 3, Rings 7. Groups 3 Definition and Examples of Rings 3.2. Basic Properties of Rings 74. Definitions and Examples of Groups 7.2. Basic Properties of Groups 7.3. Subgroups 3.3. Isomorphism 7.8. The Structure of Finite Groups 7.9. The Symmetric Group xxi CHAPTER 1 ARITHMETIC IN Z REVISITED Algebra grew out of arithmetic and depends heavily on it. So we begin our study of abstract algebra with a review of those facts from arithmetic that are used frequently in the rest of this book and provide a model for much of the work we do, We stress primarily the underlying pattern and properties rather than methods of computation. Nevertheless, the fundamental con- cepts are ones that you have seen before. 1.1 The Division Algorithm Our starting point is the set ofallintegersZ = (0,+1,%2,.. ). We assume that you are familiar with the arithmetic of integers and with the usual order relation (<) on the set Z. We also assume the WELL ORDERINGAXIOM Every nonempty subset ofthe set of nonnegative integers contains a smallest element. If you think of the nonnegative integers laid out on the usual number line, itis intuitively plausíble that each subset contains an element that lies to the left of all the other elements in the subset — this is the smallest element. On the other hand, the Well Ordering Axiom does not hold in the set Z of allintegers (there is no smallest negative integer). Nor does it hold 1.1 The Division Algorithm diviston is just repeated subtraction. For example, the first step in the long division of 4509 by 31 on page 2 amounts to subtracting 4509 — 31 - 100 = 4509 — 3100 — 1409. (This step is abbreviated by writing 45 — 31 = 14, placing a 1 in the hundreds place of the quotient, and “bringing down” a O from the divi- dend.) Similarly, the second step is subtracting 1409 — 31 - 40 = 1409 — 1240 = 169 and placing a 4 in the tens place of the quotient. These subtractions con- tinue until you reach a nonnegative number less than 31 (in this case, 14). This number is the remainder, and the number of multiples of 31 (namely, 145) that were subtracted is the quotient. So the division is really this subtraction: 4509 — 31 - 145 = 14, In the proof of the theorem, where a is being divided by b, we will subtract multiples of b from a. In other words, we will consider numbers of the form « — bx, where x is any integer (in the example, these were the numbers 4509 — 31x for various x). The smallest such nonnegative number is the remainder; the corresponding value of x is the quotient. Proofof Theorem 1.1* Leta, b, be fixedintegers, with b > 0. Consider all integers of the form a — bx, where x € Z. We first show that some of these integers must be nonnegative. There are two possibilities: 1. Waz>0,thena-b:-0=a=0.80a — bxis nonegative for x = 0 in this case. 9, [fe < 0, then —a > 0, Since b is a positive integer, we must have b =. Multiplying this last inequality by the positive number — a shows that b(—a) = —a, or equivalently, a— ba 0. Soa — bx is nonnegative when x = a in this case, Therefore the set Sof all nonnegative integers of the form « — bx, withx € Z, is nonempty. By the Well-Ordering Axiom, $ contains a smallest element — callitr. SinceresS,risofthe form a — bx for some x, say x = g. Thus we have found integers q and r such that r=a-—bga, or equivalently, a=bq+tr. Since reS, we know that r = 0. We now show that r 0 and r—>n=0 (since r="), and so q; — q must be a nonnegative integer. Therefore r — r is one of Ob, 1b, 2b, 3b, etc. But0 =r, sr 0, wemusthaveg—g=0,sothatg=a. A similar argument with the roles of r and r; reversed proves the uniqueness of q and r in the case r; = 1 and completes the proof of the theorem. E* A version of the Division Algorithm also holds when the divisor is negative: COROLIARY 1.2 Leta and c be integers with + O. Then there exist unique integers q and r such that a=cg+tr and Osr
