A concrete approach to ABSTRACT ALGEBRA by jeffrey bergen

A Concrete Approach to
Abstract Algebra


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A Concrete Approach to
Abstract Algebra
From the Integers to the Insolvability of the Quintic
Jeffrey Bergen
DePaul University
Chicago, Illinois

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Library of Congress Cataloging-in-Publication Data
Bergen, Jeffrey, 1955.
A concrete approach to abstract algebra : from the integers to the insolvability of the quintic / Jeffrey Bergen.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-374941-3 (hard cover : alk. paper) 1. Algebra, Abstract. I. Title.
QA162.B45 2010
512’.02–dc22
2009035349
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To Donna


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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
A User’s Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xix
Chapter 1 What This Book Is about and Who This Book Is for . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Finding Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Existence of Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Ruler and Compass Constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Rational Values of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Precalculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Recognizing Polynomials Using Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
1.5.1 Partial Fraction Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.2 Detecting Multiple Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Chapter 2 Proof and Intuition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
2.1 The Well Ordering Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
2.2 Proof by Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
2.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
Mathematical Induction—First Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30
Mathematical Induction—First Version Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . .32
Mathematical Induction—Second Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
Exercises for Sections 2.1, 2.2, and 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
2.4 Functions and Binary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
Exercises for Section 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

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Chapter 3 The Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
3.1 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
3.2 Unique Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
3.3 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67
Exercises for Sections 3.1, 3.2, and 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
3.4 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76
3.5 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
Exercises for Sections 3.4 and 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
Chapter 4 The Rational Numbers and the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .97
4.1 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97
4.2 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Exercises for Sections 4.1 and 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Equivalence Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Exercises for Section 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Chapter 5 The Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Fields and Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Exercises for Sections 5.1 and 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.3 Complex Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.4 Automorphisms and Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Exercises for Sections 5.3 and 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.5 Groups of Automorphisms of Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . 177
Exercises for Section 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Chapter 6 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.1 Representing Real Numbers and Complex Numbers Geometrically . . . . . . 189
6.2 Rectangular and Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Exercises for Sections 6.1 and 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.3 Demoivre’s Theorem and Roots of Complex Numbers . . . . . . . . . . . . . . . . . . . . 208
6.4 A Proof of the Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Exercises for Sections 6.3 and 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Chapter 7 The Integers Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.1 Definitions and Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.2 Zero Divisors and Invertible Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Exercises for Sections 7.1 and 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.3 The Euler φ Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.4 Polynomials with Coefficients in Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Exercises for Sections 7.3 and 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
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Chapter 8 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
8.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

I. Commutative Rings and Fields under Addition . . . . . . . . . . . . . . . . . . . . . . . . 266
II. Invertible Elements in Commutative Rings under Multiplication . . . . . 266
III. Bijections of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Exercises for Section 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
8.2 Theorems of Lagrange and Sylow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Exercises for Section 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
8.3 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Exercises for Section 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
8.4 Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Exercises for Section 8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Chapter 9 Polynomials over the Integers and Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
9.1 Integral Domains and Homomorphisms of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Exercises for Section 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
9.2 Rational Root Test and Irreducible Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Exercises for Section 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
9.3 Gauss’ Lemma and Eisenstein’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
Exercises for Section 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
9.4 Reduction Modulo p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Exercises for Section 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Chapter 10 Roots of Polynomials of Degree Less than 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 411
10.1 Finding Roots of Polynomials of Small Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
10.2 A Brief Look at Some Consequences of Galois’ Work . . . . . . . . . . . . . . . . . . . . . 418
Exercises for Sections 10.1 and 10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Chapter 11 Rational Values of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
11.1 Values of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Exercises for Section 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
Chapter 12 Polynomials over Arbitrary Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
12.1 Similarities between Polynomials and Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
12.2 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
Exercises for Sections 12.1 and 12.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

12.3 Irreducible and Minimum Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
12.4 Euclidean Algorithm and Greatest Common Divisors. . . . . . . . . . . . . . . . . . . . . . 460
Exercises for Sections 12.3 and 12.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
12.5 Formal Derivatives and Multiple Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
Exercises for Section 12.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
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Chapter 13 Difference Functions and Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
13.1 Difference Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
13.2 Polynomials and Mathematical Induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
Exercises for Sections 13.1 and 13.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
13.3 Partial Fraction Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
Exercises for Section 13.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
Chapter 14 An Introduction to Linear Algebra and Vector Spaces . . . . . . . . . . . . . . . . . 527
14.1 Examples, Examples, Examples, and a Definition. . . . . . . . . . . . . . . . . . . . . . . . . . 527
Exercises for Section 14.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
14.2 Spanning Sets and Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
14.3 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
Exercises for Sections 14.2 and 14.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
14.4 Subspaces and Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
Exercises for Section 14.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
Chapter 15 Degrees and Galois Groups of Field Extensions . . . . . . . . . . . . . . . . . . . . . . . 573
15.1 Degrees of Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Exercises for Section 15.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
15.2 Simple Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
15.3 Splitting Fields and Their Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
Exercises for Sections 15.2 and 15.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

Chapter 16 Geometric Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
16.1 Constructible Points and Constructible Real Numbers . . . . . . . . . . . . . . . . . . . . . 623
16.2 The Impossibility of Trisecting Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
Exercises for Sections 16.1 and 16.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
Chapter 17 Insolvability of the Quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
17.1 Radical Extensions and Their Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
17.2 A Proof of the Insolvability of the Quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
Exercises for Sections 17.1 and 17.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660
17.3 Kronecker’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
Exercises for Section 17.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

x


Preface
Abstract algebra, perhaps more than any other subject studied in college, has strong ties to the
mathematics courses students have taken in high school. A course in abstract algebra can
provide answers to many questions that are posed but not answered in high school
mathematics courses. This is one reason that all mathematics majors, especially those hoping
to teach at the high school or college level, can benefit from a course in abstract algebra.
Many instructors have witnessed students who, despite having had success in courses up
through multivariable calculus and linear algebra, struggle in abstract algebra. Some of these
instructors wonder if abstract algebra should even be required for math majors with a
secondary education concentration. While writing this book, I was keenly aware of these
issues.
This book was written because of my conviction that all mathematics majors should take
abstract algebra, and, more importantly, all mathematics majors can learn abstract algebra.
Some of the features that I believe will assist students in learning this subject are:

1. Links to previous mathematics courses: This book uses abstract algebra to answer basic
questions that arise in courses in algebra, geometry, trigonometry, precalculus, and
calculus. Concepts in abstract algebra are introduced as the tools needed to solve these
basic questions.
2. Exercises: Courses up through multivariable calculus and linear algebra provide students
with many exercises that allow them to practice and master new concepts. This book has
1996 exercises, many of which give students lots of practice working with concrete
examples of new concepts. For example, in Chapter 8, students will have many chances to
look at a multiplication table of a group and then compute cyclic subgroups, cosets, and
centralizers.
At various points, the exercises may appear to be somewhat repetitive. This is deliberate.
In many books, instructors find an interesting exercise and then are faced with the choice
of whether to include it in the lecture or in the homework. Perhaps the exercise’s solution
is in the solutions manual and the instructor prefers to assign problems where the solution
is not readily available. Sometimes the exact opposite situation occurs. To avoid these

xi


Preface
problems, this book often includes many similar-looking exercises. This also gives the
student more chances to practice and master concepts than is typically found in abstract
algebra texts.
3. Examples before definitions: Students in abstract algebra courses are often overwhelmed
or intimidated by the sheer volume of definitions and new objects. Whenever possible, this
book attempts to provide examples before definitions so that definitions reflect the
collecting of properties common to several concrete examples. For example, the integers,
rational numbers, real numbers, and complex numbers are introduced before the
definitions of commutative rings and fields are given. Similarly, concrete objects such as
the invertible elements of the integers modulo n and the bijections of a set are studied

before our formal discussion of groups. When new concepts are introduced, such as
automorphisms in Chapter 5 and ring homomorphisms in Chapter 9, they are immediately
applied to familiar problems such as finding roots of polynomials and determining when
polynomials are irreducible.
4. Fundamental Theorem of Algebra: Virtually every abstract algebra textbook mentions the
Fundamental Theorem of Algebra, but very few contain a proof. The reason is that a
primarily algebraic proof requires so many new ideas that it would take most books too far
off course. However, in Chapter 6, we present a proof based on some familiar ideas from
one and two variable calculus. We have chosen this direction for both philosophical and
practical reasons.
One of the goals of this book is to help students develop a deep understanding of the roots
and factoring of polynomials over different number systems. Occasionally, this requires
examining topics that are not traditionally part of an algebra course, such as the
Intermediate Value Theorem and the Fundamental Theorem of Algebra. However, these
topics are essential for an understanding of the differences in the behavior of polynomials
over the rational numbers, the real numbers, and the complex numbers.
As a practical matter, having the Fundamental Theorem of Algebra at our disposal makes
it much easier to introduce Galois theory and then prove the insolvability of the quintic.
Students often struggle with the level of abstraction in Galois theory. However, when
examining the roots of polynomials with rational coefficients, the Fundamental Theorem
of Algebra allows us to always work with fields that lie between the rational numbers and
complex numbers. This more concrete approach makes the key ideas of Galois theory
easier to understand and greatly simplifies the proof of the insolvability of the quintic.
5. Theorems with proofs: With the occasional exception of results from courses below
abstract algebra, if a theorem appears in this book, so will its proof. A philosophy
underlying this book is that reading proofs is an essential part of abstract algebra.
Sometimes textbooks will state powerful theorems, without proof, and then use them to
obtain other important results. For example, abstract algebra books often state, without
xii

Preface
proof, the Fundamental Theorem of Algebra or the Fundamental Theorem of Galois
theory and then use them to prove other results. I believe this approach can stand in the
way of students gaining a deep understanding and appreciation of algebra.
There will be times in this book when the theorems we state, prove, and apply are not the most
general results known. However, as opposed to applying stronger results whose proofs they
have never seen, I believe students will learn more applying results whose proofs they have
worked through.
Please feel free to e-mail your thoughts, comments, and corrections to me at
You can find a list of corrections at www.depaul.edu/∼jbergen.

xiii


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A User’s Guide
A yearlong course in abstract algebra can cover this entire book with sufficient time for a
thorough treatment of each section. However, it can easily be adapted to courses that meet for
only one quarter, one semester, or two quarters. For courses that run less than a year, the
chapter summaries following should help instructors decide which sections to skip and how to
sequence the sections that are covered.
Chapter 1—This introductory chapter points out that many questions that arose and were left
unanswered in a student’s previous courses in algebra, geometry, trigonometry, precalculus,
and calculus can now be answered using abstract algebra. It previews many of the results that
will be proven in this text, such as the insolvability of the quintic, the Fundamental Theorem
of Algebra, the impossibility of trisecting angles, rational values of trigonometric functions,
partial fraction decomposition, and multiple roots of polynomials. This chapter can either be

covered in class or left as a reading assignment. It is not a prerequisite for any of the later
chapters.
Chapter 2—Sections 2.1, 2.2, and 2.3 begin by discussing the importance of both intuition
and rigor in mathematics. They then focus on proofs by contradiction, the Well Ordering
Principle, and Mathematical Induction. Throughout this book, it will be very important for
your students to have a solid understanding of these sections. However, if your students are
already adept at writing proofs, these sections can be left as a reading assignment.
Section 2.4 introduces functions and binary operations. This section will be the foundation for
much of the material in this book. To make our detailed examination of groups in Chapter 8
more accessible to students, examples of groups will appear at various points before then. In
particular, groups are briefly discussed in Section 2.4 when we look at injective, surjective,
and bijective functions.
Chapter 3—This chapter focuses on properties of the integers, such as prime numbers and the
Euclidean Algorithm. The most important result in this chapter is the existence and uniqueness
of prime factorization. Exercises 31–37, immediately after Section 3.3, might be particularly
helpful to students who wonder why a concept as intuitive as unique factorization requires
proof. The ideas presented in this chapter are used throughout this book. In particular, our
discussion of polynomial rings in Chapter 12 follows the pattern set forth in this chapter.
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A User’s Guide
Chapter 4—Sections 4.1 and 4.2 contain topics that are not required for later chapters.
Section 4.1 examines rational numbers and the relationship between fractions and repeating
decimals. Section 4.2 compares the rational numbers and the real numbers and focuses on the
least upper bound property, the Intermediate Value Theorem, and roots of polynomials with
real coefficients. Some instructors may choose to skip these sections, as they cover topics that
rarely appear in abstract algebra courses. However, if a student has not seen these topics in
previous courses, they have the opportunity to see them here.
Equivalence relations and equivalence classes are introduced in Section 4.3. These topics will

reappear many times throughout this book. Since students often struggle with quotient groups
and quotient rings, many examples and exercises are provided that examine the addition and
multiplication of equivalence classes and when these operations are well defined.
Chapter 5—This chapter introduces the complex numbers and uses them, along with the
integers, rational numbers, and real numbers, to motivate the definitions of commutative
rings and fields. Complex conjugation and its relationship to roots of polynomials are then
used to motivate the definitions of automorphisms and Galois groups. Chapters 8, 15, and 17
contain a more detailed and theoretical treatment of groups and automorphisms. However, it is
helpful for students to gain experience, at this stage, working with concrete examples of these
objects.
Chapter 6—One of the themes of Chapters 5 and 6 is to demystify the complex numbers and
to show that they are as real as the real numbers. In Chapter 5, we show that the construction
of the complex numbers from the real numbers is simpler and more straightforward than either
the construction of the rational numbers from the integers or the real numbers from the rational
numbers. In Sections 6.1, 6.2, and 6.3, polar form and DeMoivre’s Theorem are introduced
and are used to help show that the addition and multiplication of complex numbers can be
viewed in a very concrete and geometric manner.
Section 6.4 contains a proof of the Fundamental Theorem of Algebra. This allows us to deal
with fields, Galois groups, and the insolvability of the quintic more concretely in Chapters 15
and 17, as we only need to work with fields that are contained in the complex numbers.
Abstract algebra courses that do not run for a full year might need to omit Chapters 15 and 17.
In this case, Section 6.4 can also be omitted.
Chapter 7—Sections 7.1, 7.2, and 7.4 examine the integers modulo n and provide many
examples of commutative rings, fields, and groups. The ideas in these sections are needed
when we examine polynomials with integer and rational coefficients in Chapter 9 and also for
the proof of Kronecker’s Theorem in Chapter 17. Section 7.3 looks at the Euler φ function and
is not a prerequisite for any of the later chapters.
Chapter 8—This chapter, which examines the structure of finite groups, can be covered in
many different ways depending on how the instructor structures the course. Since students will
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A User’s Guide
have already worked with examples of groups in Chapters 2, 5, and 7, they should be well
prepared for the more formal and detailed treatment in Chapter 8. If a course proceeds
sequentially through this text, Sections 8.1 and 8.2 will be covered toward the end of the first
semester. Therefore, even if students only take one semester of abstract algebra, they can still
see a proof of Sylow’s Theorem.
Sections 8.3 and 8.4 deal with solvable and symmetric groups and are only needed for
Chapters 15 and 17. Since Chapter 8 is quite long, instructors may decide to take a short break
from group theory after Section 8.2, as Section 8.3 can be covered at any point before
Section 15.3 and Section 8.4 at any point before Chapter 17.
If an abstract algebra course runs for only one quarter, one semester, or two quarters, the
instructor may determine that the brief introduction to groups in Chapters 2, 5, and 7 is
sufficient and then skip Chapter 8 entirely. This would allow time to cover some of the links
between abstract algebra and the high school curriculum in Chapters 9, 11, and 13 that do not
require group theory.
Chapter 9—This chapter helps to illustrate the importance of ring homomorphisms and the
integers modulo p by using them to prove the Rational Root Test, Gauss’ Lemma, and
Eisenstein’s Criterion. Since this chapter examines the roots and irreducibility of polynomials
over the integers, rationals, reals, and complex numbers, it should be particularly useful for
students planning to teach algebra at the high school or community college level.
Chapter 10—Section 10.1 shows how to find the roots of polynomials of degrees less than 5,
and Section 10.2 informally discusses some consequences of Galois’ work. Section 10.1 can
be covered at any point in the course, and Section 10.2 only requires an understanding of
Eisenstein’s Criterion. The material in this chapter is not a prerequisite for any of the later
chapters.
Chapter 11—This chapter examines rational values of trigonometric functions and explains
why the 30◦ –60◦ –90◦ and 45◦ –45◦ –90◦ triangles tend to be the only right triangles studied in
trigonometry classes. The only background material needed for this chapter is Mathematical

Induction and the Rational Root Test. This is another chapter that should be particularly useful
for future teachers. The material in this chapter is not a prerequisite for any of the later
chapters.
Chapter 12—In this chapter, it is shown that polynomials over fields satisfy analogs of many
properties satisfied by the integers. The proofs in Sections 12.1–12.4 are very similar to those
in Chapter 3. In Section 12.5, the relationship between multiple roots of polynomials and
derivatives is examined. The results in this chapter will be used repeatedly throughout the
remainder of the book.

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A User’s Guide
Chapter 13—This chapter contains material that should be of particular interest to teachers of
precalculus and calculus. In Section 13.1, difference functions are used to find the polynomial
of smallest degree that can produce a collection of data. As an application, Section 13.2 shows
how to derive many of the formulas that students merely verify when first learning about
Mathematical Induction. Section 13.3 shows why the partial fraction decomposition algorithm
in calculus courses actually works. This section relies heavily on the division algorithm and
Euclidean Algorithm for polynomial rings in Chapter 12. The material in this chapter is not a
prerequisite for any of the later chapters.
Chapter 14—This chapter examines some of the key concepts in linear algebra: basis,
dimension, spanning set, and linear independence. The material in Sections 14.1, 14.2, and
14.3 is essential for the final three chapters of this book. However, instructors may choose to
skip this chapter if the students have already taken a course in linear algebra.
Chapter 15—Section 15.1 examines degrees of field extensions, and Sections 15.2 and 15.3
look at splitting fields and Galois groups. The material in this chapter is the foundation for the
work in Chapter 16 on ruler and compass constructions and in Chapter 17 on the insolvability
of the quintic. If a course does not allow time for a proof of the insolvability of the quintic,
instructors can go directly from Section 15.1 to Chapter 16 and can also skip Sections 8.3

and 8.4.
Chapter 16—This chapter contains the proof that angles cannot be trisected with ruler and
compass. It relies very heavily on Section 15.1. Although this result appears near the end of
the book, by carefully choosing which sections to skip, it can be covered in a one-semester
course. The results in this chapter are not used in Chapter 17.
Chapter 17—Sections 17.1 and 17.2 contain the proof of the insolvability of the quintic and
also show how to produce infinite families of fifth- and seventh-degree polynomials that are
not solvable by radicals. Section 17.3 contains additional material, such as Kronecker’s
Theorem and the Isomorphism Theorem for Rings, that should be of particular interest for
students planning to pursue graduate study. This section exploits one of the recurring themes
of this book: the similarities between the integers and polynomials rings.

xviii


Acknowledgments
I would like to thank the many people who supported me as my class notes became a book.
First, my thanks to my DePaul colleagues Allan Berele and Stefan Catoiu for teaching from
preliminary drafts and to Susanna Epp and Lynn Narasimhan for encouraging me to take on
this project. Second, I thank Ken Price of the University of Wisconsin-Oshkosh for providing
useful feedback after using a preliminary draft. Third, my thanks to Glenn Olson of Maine
East High School for providing me with information about the connection between complex
numbers and electrical circuits.
I owe a debt of gratitude to Dan Tripamer of St. Viator High School for all his work producing
the diagrams. Lauren Schultz Yuhasz of Elsevier has been enormously helpful, and the
comments by the reviewers she found helped shape the final product. My thanks to Phil
Bugeau of Elsevier for his help in the final stages of this project. I would also like to thank the
University Research Council at DePaul University for their support.
Finally, a special thank you to my wife Donna and children Renee, Sabrina, Mark, and Melisa
for their continuous love and support.

Jeffrey Bergen
July 2009

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CHAPTER 1

What This Book Is about and
Who This Book Is for
You are about to embark on a journey. Often this journey is referred to as abstract algebra.
Others call it modern algebra, and still others simply call it algebra. But it is probably very
different from any type of algebra you have ever studied before.
When they are first introduced to this subject, many students feel quite intimidated. They feel
as if they are drowning in an unending sea of meaningless definitions. Terms like group, ring,
field, vector space, basis, dimension, homomorphism, isomorphism, and automorphism appear,
often for no apparent reason.
Almost all of us, at some point, are intimidated by a new project. Many home repair projects
have that effect on me. A walk through the aisles of a home improvement store can intimidate
me to the point where it becomes difficult to even formulate an intelligent question for a sales
clerk. The aisles and aisles of bizarre-looking devices and gadgets overwhelm me. However,
every item is there for a reason. Each one is a tool needed to solve a problem. Suddenly one
odd-looking device is exactly what I need to unclog my bathtub. Yet another is precisely what
I need to make my vacuum cleaner work again.
Abstract algebra is a subject that arose in an attempt to solve some very concrete problems. It
is likely that you have already come across many of these problems in your previous courses
as they occur very naturally in algebra, geometry, trigonometry, and calculus. However, in

those courses, these problems are usually dismissed with the comment that they are beyond
the scope of the course.
It may seem like an odd analogy, but reading through a book in abstract algebra is not all that
different from walking through the aisles of a home improvement store. All those intimidating
new terms you come across in an abstract algebra book are actually tools. They are precisely
the tools needed to finally solve many of the problems that arose but remained unsolved in
your previous courses.
In this book, you will be introduced to the basic terms, ideas, and concepts of abstract algebra.
Each of these new ideas will be presented as concretely as possible. New terms and concepts

Copyright © 2010 by Elsevier Inc. All rights reserved.

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2

Chapter 1

will be introduced as the tools needed to solve well-known problems. Each time we come
across a new abstract object, we will be equipped with both the knowledge of the problem it is
being used to solve, as well as multiple concrete examples of the object. This should help
eliminate the intimidating aspects of this subject and will allow us to understand and
appreciate both the beauty and the importance of the subject.
Let us now look at some of the problems that we will use abstract algebra to solve. We will list
them according to the course where you may have first seen them.

1.1 Algebra
1.1.1 Finding Roots of Polynomials
Long ago, you learned that in order to find the root of the polynomial 2x + 1, we first subtract

1 from both sides of the equation
2x + 1 = 0
to obtain the equation
2x = −1,
and then divide both sides by 2 to obtain the root
1
x=− .
2
More generally, if a and b are real numbers, with a = 0, then to find the root of the polynomial
ax + b, we first subtract b from both sides of the equation
ax + b = 0
to obtain the equation
ax = −b,
and then divide both sides by a to obtain the root
b
x=− .
a
Thus, we know how to find the root of any polynomial of degree 1. Moving on to polynomials
of degree 2, any such polynomial can be written as
ax2 + bx + c,
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What This Book Is about and Who This Book Is for

3

where a, b, and c are real numbers, with a = 0. In high school, we derived the quadratic
formula that told us that the roots of ax2 + bx + c are
√

−b ± b2 − 4ac
.
x=
2a
Therefore, for polynomials of degrees 1 and 2, it was not too difficult to find formulas for their
roots. Note that these formulas were expressions involving only the coefficients of the
polynomials and that the coefficients were combined in various ways via addition, subtraction,
multiplication, division, and taking square roots. The next natural step is to look for a
formula for the roots of polynomials of degree 3. We would like to find a formula that once
again involves the coefficients. However, we would expect that, at this point, we might not
only need to take square roots but to take cube roots as well.
More generally, our goal is to find formulas for the roots of polynomials of all possible degrees
where these formulas involve only the coefficients and the coefficients are combined in
various ways via addition, subtraction, multiplication, division, and taking roots. By taking
roots, we mean square roots, cube roots, fourth roots, and so on.
In Chapter 10, we will show that such √formulas do indeed exist for polynomials of degrees 3
2
and 4. The quadratic formula x = −b± 2ab −4ac is significantly more complicated than the
formula x = − ba for the root of polynomials of degree 1. In light of this, it is not surprising
that the formula for the roots of polynomials of degree 3 is significantly more complicated
than the quadratic formula. Again, it is no surprise that the formula for the roots of
polynomials of degree 4 is significantly more complicated than its predecessors.
The logical next step is to move on to polynomials of degree 5. Unfortunately, if one tries to
generalize or adapt the techniques used to find the roots of polynomials of degrees 1, 2, 3,
and 4, nothing seems to work. There are two possible reasons why nothing seems to work for
polynomials of degree 5. The first possible reason is that the formula is so complicated that we
just haven’t hit upon the approach needed to find it. Since the formula for the roots of
polynomials of degree 4 is so much more complicated than its predecessors, it is logical to
assume that finding a formula for the roots of polynomials of degree 5 should be an extremely
difficult task. However, there is another possible reason why we have been unsuccessful.

Perhaps there is no formula for the roots of polynomials of degree 5. This seems to be a
disturbing possibility. Not only would it be disappointing to not have a formula available for
finding the roots of polynomials of degree 5, but we also need to ask ourselves how can one
possibly prove that no such formula exists. After all, how do we prove that something can’t be
done or doesn’t exist?
In one of the greatest achievements in abstract algebra, it was shown by Galois that no formula
exists for finding the roots of polynomials of degree 5. In fact, Galois showed that for any
integer n ≥ 5, there is no formula for finding the roots of polynomials of degree n. Once again,
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4

Chapter 1

by a “formula” we mean an expression involving only the coefficients of the polynomial
where the coefficients are combined in various ways via addition, subtraction, multiplication,
division, and taking roots. This famous problem is known as the insolvability of the quintic.
Its solution will require an enormous amount of mathematical machinery and appears in
Chapter 17. The main tools needed to solve it will be group theory and Galois theory. In fact,
many of the terms and concepts appearing in this book are included because they are the tools
needed to solve this famous problem.
Before leaving this particular topic, we must remember that there are other approaches to
finding the roots of polynomials. Essentially, the insolvability of the quintic tells us that a
purely algebraic approach comes up short in trying to find the roots of some polynomials.
However, depending on the application you have in mind, you may not need a formula for
the roots of a polynomial that involves various combinations of the coefficients. Instead, you
may need the roots computed to a certain number of decimal places. There are many numerical
algorithms that can give you the roots of polynomials to as many decimal places of accuracy
as you desire (or at least as many decimal places as the machine you are using can handle).

Many of these algorithms are built into or can be easily programmed into a graphing
calculator. Although this does not technically give you the exact answer, having the answer
correct to a large number of decimal places may well be sufficient for the application you
have in mind.

1.1.2 Existence of Roots of Polynomials
As mentioned in the preceding paragraph, the phrase “finding a root” can have slightly
different meanings, depending on the context. If we are looking for the largest root of the
4
2
polynomial
√ the
√x −
√14x + 9, then in an algebra course, you would probably write
√ answer in
the form 2 + 5. (At this point, you should take a moment to check that 2 + 5 is indeed
a root of x4 − 14x2 + 9.) However, depending on the application you had in mind, you might
want the answer to 5 decimal places, and, in this case, 3.65028 would be your answer. If you
wanted the answer to 10 decimal places, then 3.6502815398 would be the answer. On the other
hand, in the unlikely event that you needed the answer to 32 decimal places, then the answer
would be
3.65028153987288474521086239294097.
Similarly, the phrase “existence of a root” can mean different things depending on the context.
We begin by considering the polynomial x + 5; if we restrict ourselves to dealing only with
positive integers, then this polynomial has no roots. Once we expand our horizons to the set of
integers, we see that this polynomial certainly has a root and the root is −5. In a similar vein,
if we restrict ourselves to dealing only with integers, then the polynomial 2x − 7 has no roots.
By once again expanding our horizons, this time to the set of rational numbers, then our
polynomial certainly has a root and the root is 72 .
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