ISBN-10:
0123749417
ISBN-13:
9780123749413
Pub. Date:
01/28/2010
Publisher:
Elsevier Science
A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic

A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic

by Jeffrey Bergen
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Overview

A Concrete Approach to Abstract Algebra presents a solid and highly accessible introduction to abstract algebra by providing details on the building blocks of abstract algebra.

It begins with a concrete and thorough examination of familiar objects such as integers, rational numbers, real numbers, complex numbers, complex conjugation, and polynomials. The author then builds upon these familiar objects and uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. Exercises provide a balanced blend of difficulty levels, while the quantity allows the instructor a latitude of choices. The final four chapters present the more theoretical material needed for graduate study.

This text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics which arise in courses in algebra, geometry, trigonometry, precalculus, and calculus.




  • Presents a more natural 'rings first' approachto effectively leading the student into the the abstract material of the course by the use of motivating concepts from previous math courses to guide the discussion of abstract algebra
  • Bridges the gap for students by showing how most of the concepts within an abstract algebra course are actually tools used to solve difficult, but well-known problems
  • Builds on relatively familiar material (Integers, polynomials) and moves onto more abstract topics, while providing a historical approach of introducing groups first as automorphisms
  • Exercises provide a balanced blend of difficulty levels, while the quantity allows the instructor a latitude of choices

Product Details

ISBN-13: 9780123749413
Publisher: Elsevier Science
Publication date: 01/28/2010
Edition description: New Edition
Pages: 720
Sales rank: 551,808
Product dimensions: 7.60(w) x 9.30(h) x 1.40(d)

About the Author

Jeffrey Bergen (DePaul, Chicago), received his B.S. in Mathematics from Brooklyn College in 1976. He received his M.S. in 1977 and Ph.D. in 1981 from the University of Chicago. His DePaul career began in 1981, where he continues to do research in the branch of abstract algebra known as noncommutative ring theory. His research has received external support from the English Speaking Union, the National Science Foundation, and the National Security Agency. He has given lectures in 7 countries and co-authored papers with 16 mathematicians around the world. In 2001, he received the Excellence in Teaching Award from the College of Liberal Arts and Sciences and, in 2007, received their Cortelyou-Lowery Award for Excellence.

Table of 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

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 737

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

Chapter 8 Croup 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

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

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