《ALGEBRA》

CHAPTERⅠ Sets，Functions，and Integers1

1.Sets1

2.Functions4

3.Relations and Binary Operations10

4.The Natural Numbers15

5.Addition and Multiplication18

6.Inequalities20

7.The Integers23

8.The Integers Modulo n28

9.Equivalence Relations and Quotient Sets33

10.Morphisms37

11.Semigroups and Monoids39

CHAPTERⅡ Groups43

1.Groups and Symmetry43

2.Rules of Calculation47

3.Cyclic Groups51

4.Subgroups56

5.Defining Relations59

6.Symmetric and Alternating Groups63

7.Transformation Groups68

8.Cosets72

9.Kernel and Image75

10.Quotient Groups79

CHAPTERⅢ Rings85

1.Axioms for Rings85

2.Constructions for Rings90

3.Quotient Rings95

4.Integral Domains and Fields99

5.The Field of Quotients101

6.Polynomials104

7.Polynomials as Functions109

8.The Division Algorithm111

9.Principal Ideal Domains115

10.Unique Factorization116

11.Prime Fields120

12.The Euclidean Algorithm122

13.Commutative Quotient Rings124

CHAPTERⅣUniversal Constructions129

1.Examples of Universals129

2.Functors131

3.Universal Elements134

4.Polynomials in Several Variables137

5.Categories141

6.Posets and Lattices143

7.Contravariance and Duality146

8.The Category of Sets153

9.The Category of Finite Sets156

CHAPTERⅤModules160

1.Sample Modules160

2.Linear Transformations163

3.Submodules167

4.Quotient Modules171

5.Free Modules173

6.Biproducts178

7.Dual Modules185

CHAPTERⅥVector Spaces193

1.Bases and Coordinates194

2.Dimension199

3.Constructions for Bases202

4.Dually Paired Vector Spaces207

5.Elementary Operations212

6.Systems of Linear Equations219

CHAPTERⅦMatrices223

1.Matrices and Free Modules224

2.Matrices and Biproducts232

3.The Matrix of a Map236

4.The Matrix of a Composite240

5.Ranks of Matrices244

6.Invertible Matrices246

7.Change of Bases251

8.Eigenvectors and Eigenvalues257

CHAPTERⅧSpecial Fields261

1.Ordered Domains261

2.The Ordered Field Q265

3.Polynomial Equations267

4.Convergence in Ordered Fields269

5.The Real Field R271

6.Polynomials over P274

7.The Complex Plane276

8.The Quaternions281

9.Extended Formal Power Series284

10.Valuations and p-adic Numbers286

CHAPTERⅨDeterminants and Tensor Products293

1.Multilinear and Alternating Functions293

2.Determinants of Matrices296

3.Cofactors and Cramer’s Rule301

4.Determinants of Maps305

5.The Characteristic Polynomial309

6.The Minimal Polynomial312

7.Universal Bilinear Functions318

8.Tensor Products319

9.Exact Sequences326

10.Identities on Tensor Products329

11.Change of Rings331

12.Algebr334

CHAPTERⅩBilinear and Quadratic Forms338

1.Bilinear Forms338

2.Symmetric Matrices341

3.Quadratic Forms343

4.Real Quadratic Forms347

5.Inner Products351

6.Ortbonormal Bases355

7.Orthogonal Matrices360

8.The Principal Axis Theorem364

9.Unitaru Spaces369

10.Normal Matrices374

CHAPTERⅪSimilar Matrices and Finite Abelian Groups378

1.Noetherian Modules378

2.Cyclic Modules381

3.Torsion Modules383

4.The Rational Canonical Form for Matrices388

5.Primary Modules392

6.Free Modules397

7.Equivalence of Matrices400

8.The Calculation of Invariant Factors404

CHAPTERⅫStructure of Groups409

1.Isomorphism Theorems409

2.Group Extensions413

3.Characteristic Subgroups417

4.Conjugate Classes419

5.The Sylow Theorems422

6.Nilpotent Groups426

7.Solvable Groups428

8.The Jordan-Holder Theorem430

9.Simplicity of An433

CHAPTERⅩⅢGalois Theory436

1.Quadratic and Cubic Equations436

2.Algebraic and Transcendental Elements439

3.Degrees442

4.Ruler and Compass445

5.Splitting Fields446

6.Galois Groups of Polynomials450

7.Separable Polynomials453

8.Finite Fields456

9.Norma！Extensions458

10.The Fundamental Theorem462

11.The Solution of Equations by Radicals465

CHAPTERⅩⅣLattices470

1.Posets：Duality Principle470

2.Lattice Identities473

3.Sublattices and Products of Lattices476

4.Modular Lattices478

5.Jordan-Holder-Dedekind Theorem480

6.Distributive Lattices483

7.Rings of Sets485

8.Boolean Algebras487

9.Free Boolean Algebras491

CHAPTERⅩⅤCategories and Adjoint Functors495

1.Categories495

2.Functors501

3.Contravariant Functors504

4.Natural Transformations506

5.Representable Functors and Universal Elements511

6.Adjoint Functors517

CHAPTERⅩⅥMultilinear Algebra522

1.Iterated Tensor Products522

2.Spaces of Tensors524

3.Graded Modules530

4.Graded Algebras533

5.The Graded Tensor Algebra539

6.The Exterior Algebra of a Module543

7.Determinants by Exterior Algebra547

8.Subspaces by Exterior Algebra552

9.Duality in Exterior Algebra555

10.Alternating Forms and Skew-Symmetric Tensors558

Bibliography561

Index565