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Ⅰ Sets, Functions, and Integers 1

1. Sets 1

2. Functions 4

3. Relations and Binary Operations 10

4. The Natural Numbers 15

5. Addition and Multiplication 18

6. Inequalities 20

7. The Integers 23

8. The Integers Modulo n 28

9. Equivalence Relations and Quotient Sets 33

10. Morphisms 37

11. Semigroups and Monoids 39

Ⅱ Groups 43

1. Groups and Symmetry 43

2. Rules of Calculation 47

3. Cyclic Groups 51

4. Subgroups 56

5. Defining Relations 59

6. Symmetric and Alternating Groups 63

7. Transformation Groups 68

8. Cosets 72

9. Kernel and Image 75

10. Quotient Groups 79

Ⅲ Rings 85

1. Axioms for Rings 85

2. Constructions for Rings 90

3. Quotient Rings 95

4. Integral Domains and Fields 99

5. The Field of Quotients 101

6. Polynomials 104

7. Polynomials as Functions 109

8. The Division Algorithm 111

9. Principal Ideal Domains 115

10. Unique Factorization 116

11. Prime Fields 120

12. The Euclidean Algorithm 122

13. Commutative Quotient Rings 124

Ⅳ Universal Constructions 129

1. Examples of Universals 129

2. Functors 131

3. Universal Elements 134

4. Polynomials in Several Variables 137

5. Categories 141

6. Posets and Lattices 143

7. Contravariance and Duality 146

8. The Category of Sets 153

9. The Category of Finite Sets 156

Ⅴ Modules 160

1. Sample Modules 160

2. Linear Transformations 163

3. Submodules 167

4. Quotient Modules 171

5. Free Modules 173

6. Biproducts 178

7. Dual Modules 185

Ⅵ Vector Spaces 193

1. Bases and Coordinates 194

2. Dimension 199

3. Constructions for Bases 202

4. Dually Paired Vector Spaces 207

5. Elementary Operations 212

6. Systems of Linear Equations 219

Ⅶ Matrices 223

1. Matrices and Free Modules 224

2. Matrices and Biproducts 232

3. The Matrix of a Map 236

4. The Matrix of a Composite 240

5. Ranks of Matrices 244

6. Invertible Matrices 246

7. Change of Bases 251

8. Eigenvectors and Eigenvalues 257

Ⅷ Special Fields 261

1. Ordered Domains 261

2. The Ordered Field Q 265

3. Polynomial Equations 267

4. Convergence in Ordered Fields 269

5. The Real Field R 271

6. Polynomials over P 274

7. The Complex Plane 276

8. The Quaternions 281

9. Extended Formal Power Series 284

10. Valuations and p-adic Numbers 286

Ⅸ Determinants and Tensor Products 293

1. Multilinear and Alternating Functions 293

2. Determinants of Matrices 296

3. Cofactors and Cramer's Rule 301

4. Determinants of Maps 305

5. The Characteristic Polynomial 309

6. The Minimal Polynomial 312

7. Universal Bilinear Functions 318

8. Tensor Products 319

9. Exact Sequences 326

10. Identities on Tensor Products 329

11. Change of Rings 331

12. Algebr 334

Ⅹ Bilinear and Quadratic Forms 338

1. Bilinear Forms 338

2. Symmetric Matrices 341

3. Quadratic Forms 343

4. Real Quadratic Forms 347

5. Inner Products 351

6. Orthonormal Bases 355

7. Orthogonal Matrices 360

8. The Principal Axis Theorem 364

9. Unitary Spaces 369

10. Normal Matrices 374

Ⅺ Similar Matrices and Finite Abelian Groups 378

1. Noetherian Modules 378

2. Cyclic Modules 381

3. Torsion Modules 383

4. The Rational Canonical Form for Matrices 388

5. Primary Modules 392

6. Free Modules 397

7. Equivalence of Matrices 400

8. The Calculation of Invariant Factors 404

Ⅻ Structure of Groups 409

1. Isomorphism Theorems 409

2. Group Extensions 413

3. Characteristic Subgroups 417

4. Conjugate Classes 419

5. The Sylow Theorems 422

6. Nilpotent Groups 426

7. Solvable Groups 428

8. The Jordan-Holder Theorem 430

9. Simplicity of An 433

ⅩⅢ Galois Theory 436

1. Quadratic and Cubic Equations 436

2. Algebraic and Transcendental Elements 439

3. Degrees 442

4. Ruler and Compass 445

5. Splitting Fields 446

6. Galois Groups of Polynomials 450

7. Separable Polynomials 453

8. Finite Fields 456

9. Normal Extensions 458

10. The Fundamental Theorem 462

11. The Solution of Equations by Radicals 465

ⅩⅣ Lattices 470

1. Posets: Duality Principle 470

2. Lattice Identities 473

3. Sublattices and Products of Lattices 476

4. Modular Lattices 478

5. Jordan-Holder-Dedekind Theorem 480

6. Distributive Lattices 483

7. Rings of Sets 485

8. Boolean Algebras 487

9. Free Boolean Algebras 491

ⅩⅤ Categories and Adjoint Functors 495

1. Categories 495

2. Functors 501

3. Contravariant Functors 504

4. Natural Transformations 506

5. Representable Functors and Universal Elements 511

6. Adjoint Functors 517

ⅩⅥ Multilinear Algebra 522

1. Iterated Tensor Products 522

2. Spaces of Tensors 524

3. Graded Modules 530

4. Graded Algebras 533

5. The Graded Tensor Algebra 539

6. The Exterior Algebra of a Module 543

7. Determinants by Exterior Algebra 547

8. Subspaces by Exterior Algebra 552

9. Duality in Exterior Algebra 555

10. Alternating Forms and Skew-Symmetric Tensors 558

Bibliography 561

Index 565