A Course in Algebra - GBV

A Course in Algebra

E. B. Vinberg

Graduate Studies in Mathematics Volume 56

American Mathematical Society Providence, Rhode Island

Contents

Preface

ix

Chapter §1.1. §1.2. §1.3. §1.4. §1.5. §1.6. §1.7. §1.8. §1.9.

1. Algebraic Structures Introduction Abelian Groups Rings and Fields Subgroups, Subrings, and Subfields The Field of Complex Numbers Rings of Residue Classes Vector Spaces Algebras Matrix Algebras

1 1 4 7 10 12 18 23 27 30

Chapter §2.1. §2.2. §2.3. §2.4. §2.5.

2. Elements of Linear Algebra Systems of Linear Equations Basis and Dimension of a Vector Space Linear Maps Determinants Several Applications of Determinants

35 35 43 53 64 76

Chapter §3.1. §3.2. §3.3.

3. Elements of Polynomial Algebra Polynomial Algebra: Construction and Basic Properties Roots of Polynomials: General Properties Fundamental Theorem of Algebra of Complex Numbers

81 81 87 93

vi

Contents

§3.4. §3.5. §3.6. §3.7. §3.8. §3.9. §3.10.

Roots of Polynomials with Real Coefficients Factorization in Euclidean Domains Polynomials with Rational Coefficients Polynomials in Several Variables Symmetric Polynomials Cubic Equations Field of Rational Fractions

98 103 109 112 116 123 129

Chapter §4.1. §4.2. §4.3. §4.4. §4.5. §4.6.

4. Elements of Group Theory Definitions and Examples Groups in Geometry and Physics Cyclic Groups Generating Sets Cosets Homomorphisms

137 137 143 147 153 155 163

Chapter §5.1. §5.2. §5.3. §5.4. §5.5.

5. Vector Spaces Relative Position of Subspaces Linear Functions Bilinear and Quadratic Functions Euclidean Spaces Hermitian Spaces

171 171 176 179 190 197

Chapter §6.1. §6.2. §6.3. §6.4. §6.5.

6. Linear Operators 201 Matrix of a Linear Operator 201 Eigenvectors 207 Linear Operators and Bilinear Functions on Euclidean Space 212 Jordan Canonical Form 221 Functions of a Linear Operator 228

Chapter §7.1. §7.2. §7.3. §7.4. §7.5.

7. Affine and Projective Spaces Affine Spaces Convex Sets Affine Transformations and Motions Quadrics Projective Spaces

Chapter 8. Tensor Algebra

"

239 239 247 259 268 280 295

Contents

§8.1. §8.2. §8.3. §8.4. Chapter §9.1. §9.2. §9.3. §9.4. §9.5. §9.6. §9.7.

vii

Tensor Product of Vector Spaces Tensor Algebra of a Vector Space Symmetric Algebra Grassmann Algebra

295 302 308 314

9. Commutative Algebra Abelian Groups Ideals and Quotient Rings Modules over Principal Ideal Domains Noetherian Rings Algebraic Extensions Finitely Generated Algebras and Affine Algebraic Varieties Prime Factorization

325 325 337 345 352 356 367 376

Chapter 10. Groups §10.1. Direct and Semidirect Products §10.2. Commutator Subgroup §10.3. Group Actions §10.4. Sylow Theorems §10.5. Simple Groups §10.6. Galois Extensions §10.7. Fundamental Theorem of Galois Theory

385 385 392 394 400 403 407 412

Chapter 11. Linear Representations and Associative Algebras §11.1. Invariant Subspaces §11.2. Complete Reducibility of Linear Representations of Finite and Compact Groups §11.3. Finite-Dimensional Associative Algebras §11.4. Linear Representations of Finite Groups §11.5. Invariants §11.6. Division Algebras

419 419 430 434 442 452 458

Chapter 12. Lie Groups §12.1. Definition and Simple Properties of Lie Groups §12.2. The Exponential Map §12.3. Tangent Lie Algebra and the Adjoint Representation §12.4. Linear Representations of Lie Groups

471 472 478 482 487

Answers to Selected Exercises

495

viii

Contents

Bibliography

501

Index

503