AlgebraPDF|Epub|txt|kindle电子书版本网盘下载

AlgebraPDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Chapter 1 Matrix Operations1

1. The Basic Operations1

2. Row Reduction9

3. Determinants18

4. Permutation Matrices24

5. Cramer’s Rule28

EXERCISES31

Chapter 2 Groups38

1. The Definition of a Group38

2. Subgroups44

3. Isomorphisms48

4. Homomorphisms51

5. Equivalence Relations and Partitions53

6. Cosets57

7. Restriction of a Homomorphism to a Subgroup59

8. Products of Groups61

9. Modular Arithmetic64

10. Quotient Groups66

EXERCISES69

Chapter 3 Vector Spaces78

1. Real Vector Spaces78

2. Abstract Fields82

3. Bases and Dimension87

4. Computation with Bases94

5. Infinite-Dimensional Spaces100

6. Direct Sums102

EXERCISES104

Chapter 4 Linear Transformations109

1. The Dimension Formula109

2. The Matrix of a Linear Transformation111

3. Linear Operators and Eigenvectors115

4. The Characteristic Polynomial120

5. Orthogonal Matrices and Rotations123

6. Diagonalization130

7. Systems of Differential Equations133

8. The Matrix Exponential138

EXERCISES145

Chapter 5 Symmetry155

1. Symmetry of Plane Figures155

2. The Group of Motions of the Plane157

3. Finite Groups of Motions162

4. Discrete Groups of Motions166

5. Abstract Symmetry： Group Operations175

6. The Operation on Cosets178

7. The Counting Formula180

8. Permutation Representations182

9. Finite Subgroups of the Rotation Group184

EXERCISES188

Chapter 6 More Group Theory197

1. The Operations of a Group on Itself197

2. The Class Equation of the Icosahedral Group200

3. Operations on Subsets203

4. The Sylow Theorems205

5. The Groups of Order 12209

6. Computation in the Symmetric Group211

7. The Free Group217

8. Generators and Relations219

9. The Todd-Coxeter Algorithm223

EXERCISES229

Chapter 7 Bilinear Forms237

1. Definition of Bilinear Form237

2. Symmetric Forms： Orthogonality243

3. The Geometry Associated to a Positive Form247

4. Hermitian Forms249

5. The Spectral Theorem253

6. Conics and Quadrics255

7. The Spectral Theorem for Normal Operators259

8. Skew-Symmetric Forms260

9. Summary of Results， in Matrix Notation261

EXERCISES262

Chapter 8 Linear Groups270

1. The Classical Linear Groups270

2. The Special Unitary Group SU2272

3. The Orthogonal Representation of SU2276

4. The Special Linear Group SL2（R）281

5. One-Parameter Subgroups283

6. The Lie Algebra286

7. Translation in a Group292

8. Simple Groups295

EXERCISES300

Chapter 9 Group Representations307

1. Definition of a Group Representation307

2. G-Invariant Forms and Unitary Representations310

3. Compact Groups312

4. G-Invariant Subspaces and Irreducible Representations314

5. Characters316

6. Permutation Representations and the Regular Representation321

7. The Representations of the Icosahedral Group323

8. One-Dimensional Representations325

9. Schur’s Lemma， and Proof of the Orthogonality Relations325

10. Representations of the Group SU2330

EXERCISES335

Chapter 10 Rings345

1. Definition of a Ring345

2. Formal Construction of Integers and Polynomials347

3. Homomorphisms and Ideals353

4. Quotient Rings and Relations in a Ring359

5. Adjunction of Elements364

6. Integral Domains and Fraction Fields368

7. Maximal Ideals370

8. Algebraic Geometry373

EXERCISES379

Chapter 11 Factorization389

1. Factorization of Integers and Polynomials389

2. Unique Factorization Domains， Principal Ideal Domains，and Euclidean Domains392

3. Gauss’s Lemma398

4. Explicit Factorization of Polynomials402

5. Primes in the Ring of Gauss Integers406

6. Algebraic Integers409

7. Factorization in Imaginary Quadratic Fields414

8. Ideal Factorization419

9. The Relation Between Prime Ideals of R and Prime Integers424

10. Ideal Classes in Imaginary Quadratic Fields425

11. Real Quadratic Fields433

12. Some Diophantine Equations437

EXERCISES440

Chapter 12 Modules450

1. The Definition of a Module450

2. Matrices， Free Modules， and Bases452

3. The Principle of Permanence of Identities456

4. Diagonalization of Integer Matrices457

5. Generators and Relations for Modules464

6. The Structure Theorem for Abelian Groups471

7. Application to Linear Operators476

8. Free Modules over Polynomial Rings482

EXERCISES483

Chapter 13 Fields492

1. Examples of Fields492

2. Algebraic and Transcendental Elements493

3. The Degree of a Field Extension496

4. Constructions with Ruler and Compass500

5. Symbolic Adjunction of Roots506

6. Finite Fields509

7. Function Fields515

8. Transcendental Extensions525

9. Algebraically Closed Fields527

EXERCISES530

Chapter 14 Galois Theory537

1. The Main Theorem of Galois Theory537

2. Cubic Equations543

3. Symmetric Functions547

4. Primitive Elements552

5. Proof of the Main Theorem556

6. Quartic Equations560

7. Kummer Extensions565

8. Cyclotomic Extensions567

9. Quintic Equations570

EXERCISES575

Appendix Background Material585

1. Set Theory585

2. Techniques of Proof589

3. Topology593

4. The Implicit Function Theorem597

EXERCISES599

Notation601

Suggestions for Further Reading603

Index607