A Concrete Approach to Abstract Algebra

A Concrete Approach to Abstract Algebra begins with a concrete and thorough examination of familiar objects like integers, rational numbers, real numbers, complex numbers, complex conjugation and polynomials, in this unique approach, the author builds upon these familar objects and then uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. The text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics wich arise in courses in algebra, geometry, trigonometry, precalculus and calculus. The final four chapters present the more theoretical material needed for graduate study. Presents a more natural 'rings first' approach to 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

Preface	p. xi
A User's Guide	p. xv
Acknowledgments	p. xix
What This Book Is about and Who This Book Is for	p. 1
Algebra	p. 2
Finding Roots of polynomials	p. 2
Existence of Roots of Polynomials	p. 4
Solving Linear Equations	p. 5
Geometry	p. 6
Ruler and Compass Constructions	p. 6
Trigonometry	p. 7
Rational Values of Trigonometric Functions	p. 7
Precalculus	p. 8
Recognizing Polynomials Using Data	p. 8
Calculus	p. 10
Partial Fraction Decomposition	p. 10
Detecting Multiple Roots of Polynomials	p. 12
Exercises for Chapter 1	p. 14
Proof and intuition	p. 19
The Well Ordering Principle	p. 20
Proof by Contradiction	p. 26
Mathematical Induction	p. 29
Mathematical Induction-First Version	p. 30
Mathematical Induction-First Version Revisited	p. 32
Mathematical Induction-Second Version	p. 37
Exercises for Sections 2.1, 2.2, and 2.3	p. 37
Functions and Binary Operations	p. 46
Exercises for Section 2.4	p. 56
The Integers	p. 61
Prime Numbers	p. 61
Unique Factorization	p. 64
Division Algorithm	p. 67
Exercises for Sections 3.1, 3.2, and 3.3	p. 71
Greatest Common Divisors	p. 76
Euclidean Algorithm	p. 79
Exercises for Sections 3.4 and 3.5	p. 91
The Rational Numbers and the Real Numbers	p. 97
Rational Numbers	p. 97
Intermediate Value Theorem	p. 105
Exercises for Sections 4.1 and 4.2	p. 113
Equivalence Relations	p. 118
Exercises for Section 4.3	p. 128
The Complex Numbers	p. 737
Complex Numbers	p. 137
Fields and Commutative Rings	p. 140
Exercises for Sections 5.1 and 5.2	p. 148
Complex Conjugation	p. 154
Automorphisms and Roots of Polynomials	p. 163
Exercises for Sections 5.3 and 5.4	p. 169
Groups of Automorphisms of Commutative Rings	p. 177
Exercises for Section 5.5	p. 182
The Fundamental Theorem of Algebra	p. 189
Representing Real Numbers and Complex Numbers Geometrically	p. 189
Rectangular and Polar Form	p. 199
Exercises for Sections 6.1 and 6.2	p. 203
Demoivre's Theorem and Roots of Complex Numbers	p. 208
A Proof of the Fundamental Theorem of Algebra	p. 215
Exercises for Sections 6.3 and 6.4	p. 222
The Integers Modulo n	p. 227
Definitions and Basic Properties	p. 227
Zero Divisors and Invertible Elements	p. 233
Exercises for Sections 7.1 and 7.2	p. 241
The Euler ¿ Function	p. 248
Polynomials with Coefficients in Z_n	p. 256
Exercises for Sections 7.3 and 7.4	p. 260
Croup Theory	p. 265
Definitions and Examples	p. 265
Commutative Rings and Fields under Addition	p. 266
Invertible Elements in Commutative Rings under Multiplication	p. 266
Bijections of Sets	p. 267
Exercises for Section 8.1	p. 288
Theorems of Lagrange and Sylow	p. 294
Exercises for Section 8.2	p. 318
Solvable Groups	p. 322
Exercises for Section 8.3	p. 342
Symmetric Groups	p. 347
Exercises for Section 8.4	p. 361
Polynomials over the Integers and Rationals	p. 365
Integral Domains and Homomorphisms of Rings	p. 365
Exercises for Section 9.1	p. 374
Rational Root Test and Irreducible Polynomials	p. 379
Exercises for Section 9.2	p. 387
Gauss' Lemma and Eisenstein's Criterion	p. 390
Exercises for Section 9.3	p. 397
Reduction Modulo p	p. 398
Exercises for Section 9.4	p. 408
Roots of Polynomials of Degree Less than 5	p. 411
Finding Roots of Polynomials of Small Degree	p. 411
A Brief Look at Some Consequences of Galois'Work	p. 418
Exercises for Sections 10.1 and 10.2	p. 420
Rational Values of Trigonometric Functions	p. 423
Values of Trigonometric Functions	p. 424
Exercises for Section 11.1	p. 433
Polynomials over Arbitrary Fields	p. 437
Similarities between Polynomials and Integers	p. 437
Division Algorithm	p. 444
Exercises for Sections 12.1 and 12.2	p. 453
Irreducible and Minimum Polynomials	p. 457
Euclidean Algorithm and Greatest Common Divisors	p. 460
Exercises for Sections 12.3 and 12.4	p. 470
Formal Derivatives and Multiple Roots	p. 474
Exercises for Section 12.5	p. 484
Difference Functions and Partial Fractions	p. 487
Difference Functions	p. 488
Polynomials and Mathematical Induction	p. 499
Exercises for Sections 13.1 and 13.2	p. 504
Partial Fraction Decomposition	p. 510
Exercises for Section 13.3	p. 523
An Introduction to Linear Algebra and Vector Spaces	p. 527
Examples, Examples, Examples, and a Definition	p. 527
Exercises for Section 14.1	p. 538
Spanning Sets and Linear Independence	p. 540
Basis and Dimension	p. 548
Exercises for Sections 14.2 and 14.3	p. 555
Subspaces and Linear Equations	p. 560
Exercises for Section 14.4	p. 568
Degrees and Galois Groups of Field Extensions	p. 573
Degrees of Field Extensions	p. 573
Exercises for Section 15.1	p. 590
Simple Extensions	p. 594
Splitting Fields and Their Galois Groups	p. 599
Exercises for Sections 15.2 and 15.3	p. 615
Geometric Constructions	p. 623
Constructible Points and Constructible Real Numbers	p. 623
The Impossibility of Trisecting Angles	p. 639
Exercises for Sections 16.1 and 16.2	p. 643
Insolvability of the Quintic	p. 645
Radical Extensions and Their Galois Groups	p. 645
A Proof of the Insolvability of the Quintic	p. 657
Exercises for Sections 17.1 and 17.2	p. 660
Kronecker's Theorem	p. 663
Exercises for Section 17.3	p. 678
Bibliography	p. 685
Index	p. 687
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A Concrete Approach to Abstract Algebra

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