simple arithmetic

Instructions 1. You may use a calculator to help you with simple arithmetic (+, −, ×, /). No other computational aids are permitted. 2. You may consult the course textbook (Beachy/Blair) and your own notes, and you may ask me for help. No other sources (online or otherwise) are permitted. Problems 1. Let f (x) = x3 + 5×2 + 1 and let u ∈ C be a root of f (x). (a) Show that f (x) is irreducible over Q. (b) Express (1 + u2 )−1 in the form a + bu + cu2 for a, b, c ∈ Q. 2. (a) Suppose α ∈ C is of degree 5 over Q. Prove that K(α) = K(α3 ). (b) Show that every irrational element of Q(π) is transcendental. (You may assume π is transcendental.) (c) Let a, b ∈ C. Show that if both a + b and ab are algebraic, then so are a and b. √ √ 3. Let F = Q( 2, 3 3). (a) Calculate [F : Q] and give a basis of F over Q. √ √ (b) Is 2 − 3 3 a primitive element for F over Q? Explain. 4. (a) Show that cos( 2π 7 ) is algebraic over Q by finding its minimal polynomial. (Hint: α = 12 (ζ + ζ −1 ) where ζ = e2πi/7 ) (b) Show that Q(α) is a normal extension of Q. √ (c) Is there a rational number β for which Q(α) = Q( 3 β)? Explain. (d) Prove that a regular heptagon is not constructible by ruler and compass. 5. Construct the splitting field of f (x) = x3 + 2×2 + 1 over Z3 and explicitly list the elements of its Galois group. 6. Let g(x) = x5 − x4 − 4x + 4. (a) Find the splitting field F of g(x) over Q and explicitly describe G = Gal(F/Q) as a subgroup of S5 . (b) List all subgroups of G and the corresponding intermediate fields between Q and F . 7. The Galois group of f (x) ∈ Q[x] over Q is of order 10. Prove that f (x) is solvable by radicals. (Hint: Z10 and D5 are the only groups of order 10.) 8. The polynomial f (x) ∈ Q[x] is irreducible and its Galois group over Q has order 9. What can you deduce about f (x)? (Note: This is an open-ended question. Tell me whatever you can.) 1 Abstract Algebra Fourth Edition John A. Beachy William D. Blair Northern Illinois University WAVELAND PRESS, INC. Long Grove, Illinois For information about this book, contact: Waveland Press, Inc. 4180 IL Route 83, Suite 101 Long Grove, IL 60047-9580 (847) 634-0081 info@waveland.com www.waveland.com Copyright © 2019 by John A. Beachy and William D. Blair 10-digit ISBN 1-4786-3869-9 13-digit ISBN 978-1-4786-3869-8 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means without permission in writing from the publisher. Printed in the United States of America 7 6 5 4 3 2 1 Contents PREFACE vii PREFACE TO THE THIRD EDITION viii PREFACE TO THE SECOND EDITION ix TO THE STUDENT xiii WRITING PROOFS xvi HISTORICAL BACKGROUND xxi 1 2 3 INTEGERS 1.1 Divisors . . . . . . 1.2 Primes . . . . . . . 1.3 Congruences . . . 1.4 Integers Modulo n . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 16 27 38 50 FUNCTIONS 2.1 Functions . . . . . . . 2.2 Equivalence Relations . 2.3 Permutations . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 53 66 76 90 GROUPS 3.1 Definition of a Group . 3.2 Subgroups . . . . . . . 3.3 Constructing Examples 3.4 Isomorphisms . . . . . 3.5 Cyclic Groups . . . . . 3.6 Permutation Groups . . 3.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 92 108 122 132 143 150 161 . . . . . iii iv CONTENTS 3.8 Cosets, Normal Subgroups, and Factor Groups . . . . . . . . . . . 173 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4 5 6 7 8 POLYNOMIALS 4.1 Fields; Roots of Polynomials . . 4.2 Factors . . . . . . . . . . . . . . 4.3 Existence of Roots . . . . . . . 4.4 Polynomials over Z, Q, R, and C Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 189 202 213 221 234 COMMUTATIVE RINGS 5.1 Commutative Rings; Integral Domains 5.2 Ring Homomorphisms . . . . . . . . 5.3 Ideals and Factor Rings . . . . . . . . 5.4 Quotient Fields . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 236 248 262 273 280 FIELDS 6.1 Algebraic Elements . . . . . . . . . . . . 6.2 Finite and Algebraic Extensions . . . . . 6.3 Geometric Constructions . . . . . . . . . 6.4 Splitting Fields . . . . . . . . . . . . . . 6.5 Finite Fields . . . . . . . . . . . . . . . . 6.6 Irreducible Polynomials over Finite Fields 6.7 Quadratic Reciprocity . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 282 289 295 301 307 313 320 328 STRUCTURE OF GROUPS 7.1 Isomorphism Theorems; Automorphisms 7.2 Conjugacy . . . . . . . . . . . . . . . . . 7.3 Groups Acting on Sets . . . . . . . . . . 7.4 The Sylow Theorems . . . . . . . . . . . 7.5 Finite Abelian Groups . . . . . . . . . . . 7.6 Solvable Groups . . . . . . . . . . . . . . 7.7 Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 330 337 345 353 358 366 374 GALOIS THEORY 8.1 The Galois Group of a Polynomial . . . . . . 8.2 Multiplicity of Roots . . . . . . . . . . . . . 8.3 The Fundamental Theorem of Galois Theory 8.4 Solvability by Radicals . . . . . . . . . . . . 8.5 Cyclotomic Polynomials . . . . . . . . . . . 8.6 Computing Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 384 390 394 405 411 417 . . . . . . . . . . CONTENTS 9 v UNIQUE FACTORIZATION 427 9.1 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . . 428 9.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . . . 435 9.3 Some Diophantine Equations . . . . . . . . . . . . . . . . . . . . 441 10 GROUPS: SELECTED TOPICS 10.1 Nilpotent Groups . . . . . . . . . . . . 10.2 Internal Semidirect Products of Groups 10.3 External Semidirect Products of Groups 10.4 Classification of Groups of Small Order 10.5 The Orthogonal Group O2 .R/ . . . . . 10.6 Isometries of R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 453 457 463 471 477 481 APPENDIX A.1 Sets . . . . . . . . . . . . . . . . . . . A.2 Construction of the Number Systems . . A.3 Basic Properties of the Integers . . . . . A.4 Induction . . . . . . . . . . . . . . . . A.5 Complex Numbers . . . . . . . . . . . A.6 Solution of Cubic and Quartic Equations A.7 Dimension of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 487 490 493 495 498 504 512 BIBLIOGRAPHY 516 SELECTED ANSWERS 518 INDEX OF SYMBOLS 527 INDEX 531 PREFACE In this edition we hope to have corrected mistakes and typographical errors in the previous one. In the early chapters, we have added a few examples and exercises; in Chapters 7 and 8 we have added a significant number of more difficult exercises. We have also added new material on groups in Chapter 10, including semidirect products and the classification of groups of order < 16. The last two sections of the chapter are devoted to applications of group theory to the geometry of the plane. On meeting a new theorem, it is important for the student to look back at the theorems on which it depends. Perhaps just as important is to have some idea as to where it might be used later in the development. To address this, to the early chapters we have added a few paragraphs titled “Looking ahead”, to put some theorems in context. We would like to point out to both students and instructors that there is some supplementary material available on the book’s website: www:mat h:ni u:edu= beachy=abst ract algebra=. This includes solutions to a few selected exercises, marked by the symbol  in the text. Many of these exercises are ones that are referred to in the text or are used in a significant way in subsequent exercises. The file Selected Solutions for Students is also available on the Waveland Press website. In addition to the books mentioned in the prefaces of earlier editions, we wish to acknowledge the influence of the books of M. Artin and A. R. Wadsworth listed in the bibliography. Finally, we would like to thank our publisher, Neil Rowe, for his continued support, and, in particular, for his encouragement to work on another edition. DeKalb, Illinois John A. Beachy William D. Blair December 1, 2018 vii viii PREFACE PREFACE TO THE THIRD EDITION This edition would probably not have been written without the impetus from George Bergman, of the University of California, Berkeley. After using the book, on more than one occasion he sent us a large number of detailed suggestions on how to improve the presentation. Many of these were in response to questions from his students, so we owe an enormous debt of gratitude to his students, as well as to Professor Bergman. We believe that our responses to his suggestions and corrections have measurably improved the book. We would also like to acknowledge important corrections and suggestions that we received from Marie Vitulli, of the University of Oregon, and from David Doster, of Choate Rosemary Hall. We have also benefited over the years from numerous comments from our own students and from a variety of colleagues. We would like to add Doug Bowman, Dave Rusin, and Jeff Thunder to the list of colleagues given in the preface to the second edition. In this edition we have added a number of exercises, we have added 1 to all rings, and we have done our best to weed out various errors and misprints. We use the book in a linear fashion, but there are some alternatives to that approach. With students who already have some acquaintance with the material in Chapters 1 and 2, it would be possible to begin with Chapter 3, on groups, using the first two chapters for a reference. We view these chapters as studying cyclic groups and permutation groups, respectively. Since Chapter 7 continues the development of group theory, it is possible to go directly from Chapter 3 to Chapter 7. Chapter 5 contains basic facts about commutative rings, and contains many examples which depend on a knowledge of polynomial rings from Chapter 4. Chapter 5 also depends on Chapter 3, since we make use of facts about groups in the development of ring theory, particularly in Section 5.3 on factor rings. After covering Chapter 5, it is possible to go directly to Chapter 9, which has more ring theory and some applications to number theory. Our development of Galois theory in Chapter 8 depends on results from Chapters 5 and 6. Section 8.4, on solvability by radicals, requires a significant amount of material from Chapter 7. Rather than outlining a large number of possible paths through various parts of the text, we have to ask the instructor to read ahead and use a great deal of caution in choosing any paths other than the ones we have suggested above. Finally, we would like to thank our publisher, Neil Rowe, for his continued support of our writing. DeKalb, Illinois John A. Beachy William D. Blair September 1, 2005 PREFACE ix PREFACE TO THE SECOND EDITION An abstract algebra course at the junior/senior level, whether for one or two semesters, has been a well-established part of the curriculum for mathematics majors for over a generation. Our book is intended for this course, and has grown directly out of our experience in teaching the course at Northern Illinois University. As a prerequisite to the abstract algebra course, our students are required to have taken a sophomore level course in linear algebra that is largely computational, although they have been introduced to proofs to some extent. Our classes include students preparing to teach high school, but almost no computer science or engineering students. We certainly do not assume that all of our students will go on to graduate school in pure mathematics. In searching for appropriate text books, we have found several texts that start at about the same level as we do, but most of these stay at that level, and they do not teach nearly as much mathematics as we desire. On the other hand, there are several fine books that start and finish at the level of our Chapters 3 through 6, but these books tend to begin immediately with the abstract notion of group (or ring), and then leave the average student at the starting gate. We have in the past used such books, supplemented by our Chapter 1. Historically the subject of abstract algebra arose from concrete problems, and it is our feeling that by beginning with such concrete problems we will be able to generate the student’s interest in the subject and at the same time build on the foundation with which the student feels comfortable. Although the book starts in a very concrete fashion, we increase the level of sophistication as the book progresses, and, by the end of Chapter 6, all of the topics taught in our course have been covered. It is our conviction that the level of sophistication should increase, slowly at first, as the students become familiar with the subject. We think our ordering of the topics speaks directly to this assertion. Recently there has been a tendency to yield to demands of “relevancy,” and to include “applications” in this course. It is our feeling that such inclusions often tend to be superficial. In order to make room for the inclusion of applications, some important mathematical concepts have to be sacrificed. It is clear that one must have substantial experience with abstract algebra before any genuine applications can be treated. For this reason we feel that the most honest introduction concentrates on the algebra. One of the reasons frequently given for treating applications is that they motivate the student. We prefer to motivate the subject with concrete problems from areas that the students have previously encountered, namely, the integers and polynomials over the real numbers. One problem with most treatments of abstract algebra, whether they begin with group theory or ring theory, is that the students simultaneously encounter for the first time both abstract mathematics and the requirement that they produce proofs x PREFACE of their own devising. By taking a more concrete approach than is usual, we hope to separate these two initiations. In three of the first four chapters of our book we discuss familiar concrete mathematics: number theory, functions and permutations, and polynomials. Although the objects of study are concrete, and most are familiar, we cover quite a few nontrivial ideas and at the same time introduce the student to the subtle ideas of mathematical proof. (At Northern Illinois University, this course and Advanced Calculus are the traditional places for students to learn how to write proofs.) After studying Chapters 1 and 2, the students have at their disposal some of the most important examples of groups—permutation groups, the group of integers modulo n, and certain matrix groups. In Chapter 3 the abstract definition of a group is introduced, and the students encounter the notion of a group armed with a variety of concrete examples. Probably the most difficult notion in elementary group theory is that of a factor group. Again this is a case where the difficulty arises because there are, in fact, two new ideas encountered together. We have tried to separate these by treating the notions of equivalence relation and partition in Chapter 2 in the context of sets and functions. We consider there the concept of factoring a function into “better” functions, and show how the notion of a partition arises in this context. These ideas are related to the integers modulo n, studied in Chapter 1. When factor groups are introduced in Chapter 3, we have partitions and equivalence relations at our disposal, and we are able to concentrate on the group structure introduced on the equivalence classes. In Chapter 4 we return to a more concrete subject when we derive some important properties of polynomials. Here we draw heavily on the students’ familiarity with polynomials from high school algebra and on the parallel between the properties of the integers studied in Chapter 1 and the polynomials. Chapter 5 then introduces the abstract definition of a ring after we have already encountered several important examples of rings: the integers, the integers modulo n, and the ring of polynomials with coefficients in any field. From this point on our book looks more like a traditional abstract algebra textbook. After rings we consider fields, and we include a discussion of root adjunction as well as the three problems from antiquity: squaring the circle, duplicating the cube, and trisecting an angle. We also discuss splitting fields and finite fields here. We feel that the first six chapters represent the most that students at institutions such as ours can reasonably absorb in a year. Chapter 7 returns to group theory to consider several more sophisticated ideas including those needed for Galois theory, which is the subject matter of Chapter 8. In Chapter 9 we return to a study of rings, and consider questions of unique factorization. As a number theoretic application, we present a proof of Fermat’s last theorem for the exponent 3. In fact, this is the last of a thread of number theoretic applications that run through the text, including a proof of the quadratic reciprocity law in Section 6.7 and a study of primitive roots modulo p in Section 7.5. The applications to number theory provide topics suitable for honors students. PREFACE xi The last three chapters are intended to make the book suitable for an honors course or for classes of especially talented or well-prepared students. In these chapters the writing style is rather terse and demanding. Proofs are included for the Sylow theorems, the structure theorem for finite abelian groups, theorems on the simplicity of the alternating group and the special linear group over a finite field, the fundamental theorem of Galois theory, Abel’s theorem on the insolvability of the quintic, and the theorem that a polynomial ring over a unique factorization domain is again a unique factorization domain. The only prerequisite for our text is a sophomore level course in linear algebra. We do not assume that the student has been required to write, or even read, proofs before taking our course. We …
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