Algebra

MacLane S.,  G. Birkhoff. Algebra (ed.3,Chelsea,1988) ......Page 3
Copyright ......Page 4
Preface to the Third Edition ......Page 5
From the Preface to the First Edition ......Page 7
From the Preface to the Second Edition ......Page 10
Contents ......Page 11
LIST OF SYMBOLS  ......Page 16
1.Sets • 1 ......Page 19
2.Functions • 4 ......Page 22
3.Relations and Binary Operations • 10 ......Page 28
4.The Natural Numbers • 15 ......Page 33
5.Addition and Multiplication • 18 ......Page 36
6.Inequalities • 20 ......Page 38
7.The Integers • 23 ......Page 41
8.The Integers Modulo n • 28 ......Page 46
9.Equivalence Relations and Quotient Sets  33......Page 51
10.Morphisms • 37 ......Page 55
11.Semigroups and Monoids •39 ......Page 57
1.Groups and Symmetry •43 ......Page 61
2.Rules of Calculation • 47 ......Page 65
3.Cyclic Groups •51 ......Page 69
4.Subgroups • 56 ......Page 74
5.Defining Relations • 59 ......Page 77
6.Symmetric and Alternating Groups • 63 ......Page 81
7.Transformation Groups • 68 ......Page 86
8.Cosets • 72 ......Page 90
9.Kernel and Image • 75 ......Page 93
10.Quotient Groups • 79 ......Page 97
1.Axioms for Rings • 85 ......Page 103
2.Constructions for Rings • 90 ......Page 108
3.Quotient Rings • 95 ......Page 113
4.Integral Domains and Fields • 99 ......Page 117
5.The Field of Quotients • 101 ......Page 119
6.Polynomials • 104 ......Page 122
7.Polynomials as Functions • 109 ......Page 127
8.The Division Algorithm •111 ......Page 129
9.Principal Ideal Domains • 115 ......Page 133
10.Unique Factorization • 116 ......Page 134
11.Prime Fields • 120 ......Page 138
12.The Euclidean Algorithm • 122 ......Page 140
13.Commutative Quotient Rings • 124 ......Page 142
1.Examples of Universals • 129 ......Page 147
2.Functors • 131 ......Page 149
3.Universal Elements • 134 ......Page 152
4.Polynomials in Several Variables • 137 ......Page 155
5.Categories • 141 ......Page 159
6.Posets and Lattices • 143 ......Page 161
7.Contravariance and Duality • 146 ......Page 164
8.The Category of Sets • 153 ......Page 171
9.The Category of Finite Sets • 156 ......Page 174
1.Sample Modules • 160 ......Page 178
2.Linear Transformations • 163 ......Page 181
3.Submodules • 167 ......Page 185
4.Quotient Modules •171 ......Page 189
5.Free Modules • 173 ......Page 191
6.Biproducts • 178 ......Page 196
7.Dual Modules • 185 ......Page 203
CHAPTER VI Vector Spaces 193 ......Page 211
1.Bases and Coordinates • 194 ......Page 212
2.Dimension • 199 ......Page 217
3.Constructions for Bases • 202 ......Page 220
4.Dually Paired Vector Spaces • 207 ......Page 225
5.Elementary Operations • 212 ......Page 230
6.Systems of Linear Equations •219 ......Page 237
CHAPTER VII Matrices 223 ......Page 241
1.Matrices and Free Modules •224 ......Page 242
2.Matrices and Biproducts • 232 ......Page 250
3.The Matrix of a Map • 236 ......Page 254
4.The Matrix of a Composite • 240 ......Page 258
5.Ranks of Matrices • 244  ......Page 262
6.Invertible Matrices • 246 ......Page 264
7.Change of Bases • 251 ......Page 269
8.Eigenvectors and Eigenvalues • 257 ......Page 275
1.Ordered Domains • 261 ......Page 279
2.The Ordered Field Q • 265 ......Page 283
3.Polynomial Equations • 267 ......Page 285
4.Convergence in Ordered Fields • 269 ......Page 287
5.The Real Field R •271 ......Page 289
6.Polynomials over R • 274 ......Page 292
7.The Complex Plane • 276 ......Page 294
8.The Quaternions • 281 ......Page 299
9.Extended Formal Power Series • 284 ......Page 302
10.Valuations and p-adic Numbers • 286 ......Page 304
1.Multilinear and Alternating Functions • 293 ......Page 311
2.Determinants of Matrices • 296 ......Page 314
3.Cofactors and Cramer’s Rule • 301 ......Page 319
4.Determinants of Maps • 305 ......Page 323
5.The Characteristic Polynomial • 309 ......Page 327
6.The Minimal Polynomial •312 ......Page 330
7.Universal Bilinear Functions •318 ......Page 336
8.Tensor Products • 319 ......Page 337
9.Exact Sequences • 326 ......Page 344
10.Identities on Tensor Products • 329 ......Page 347
11.Change of Rings •331 ......Page 349
12.Algebras • 334 ......Page 352
1.Bilinear Forms • 338 ......Page 356
2.Symmetric Matrices • 341 ......Page 359
3.Quadratic Forms • 343 ......Page 361
4.Real Quadratic Forms • 347 ......Page 365
5.Inner Products • 351 ......Page 369
6.Orthonormal Bases • 355 ......Page 373
7.Orthogonal Matrices • 360 ......Page 378
8.The Principal Axis Theorem • 364 ......Page 382
9.Unitary Spaces • 369 ......Page 387
10.Normal Matrices • 374 ......Page 392
1.Noetherian Modules •378 ......Page 396
2.Cyclic Modules • 381 ......Page 399
3.Torsion Modules • 383 ......Page 401
4.The Rational Canonical Form for Matrices • 388 ......Page 406
5.Primary Modules • 392 ......Page 410
6.Free Modules • 397 ......Page 415
7.Equivalence of Matrices • 400 ......Page 418
8.The Calculation of Invariant Factors • 404 ......Page 422
1.Isomorphism Theorems • 409 ......Page 427
2.Group Extensions •413 ......Page 431
3.Characteristic Subgroups • 417 ......Page 435
4.Conjugate Classes •419 ......Page 437
5.The Sylow Theorems • 422 ......Page 440
6.Nilpotent Groups • 426 ......Page 444
7.Solvable Groups • 428 ......Page 446
8.The Jordan-Holder Theorem • 430 ......Page 448
9.Simplicity of An • 433 ......Page 451
1.Quadratic and Cubic Equations • 436 ......Page 454
2.Algebraic and Transcendental Elements • 439 ......Page 457
3.Degrees • 442 ......Page 460
4.Ruler and Compass • 445 ......Page 463
5.Splitting Fields • 446 ......Page 464
6.Galois Groups of Polynomials • 450 ......Page 468
7.Separable Polynomials • 453 ......Page 471
8.Finite Fields • 456 ......Page 474
9.Normal Extensions • 458 ......Page 476
10.The Fundamental Theorem • 462 ......Page 480
11.The Solution of Equations by Radicals • 465 ......Page 483
1.Posets: Duality Principle •470 ......Page 488
2.Lattice Identities • 473 ......Page 491
3.Sublattices and Products of Lattices • 476 ......Page 494
4.odular Lattices • 478 ......Page 496
5.Jordan-Holder-Dedekind Theorem • 480 ......Page 498
6.Distributive Lattices • 483 ......Page 501
7.Rings of Sets • 485 ......Page 503
8.Boolean Algebras • 487 ......Page 505
9.Free Boolean Algebras • 491 ......Page 509
1.Categories • 495 ......Page 513
2.Functors • 501 ......Page 519
3.Contravariant Functors • 504 ......Page 522
4.Natural Transformations • 506 ......Page 524
5.Representable Functors and Universal Elements •511 ......Page 529
6.Adjoint Functors •517 ......Page 535
1.Iterated Tensor Products • 522 ......Page 540
2.Spaces of Tensors • 524 ......Page 542
3.Graded Modules • 530 ......Page 548
4.Graded Algebras • 533 ......Page 551
5.The Graded Tensor Algebra •539 ......Page 557
6.The Exterior Algebra of a Module • 543 ......Page 561
7.Determinants by Exterior Algebra • 547 ......Page 565
8.Subspaces by Exterior Algebra • 552 ......Page 570
9.Duality in Exterior Algebra • 555 ......Page 573
10.Alternating Forms and Skew-Symmetric Tensors • 558 ......Page 576
1.The Affine Line • 561 ......Page 579
2.Affine Spaces • 564 ......Page 582
3.The Affine Group • 570 ......Page 588
4.Affine Subspaces • 576 ......Page 594
5.Biaffine and Quadratic Functionals • 578 ......Page 596
6.Euclidean Spaces • 586 ......Page 604
7.Euclidean Quadrics • 590 ......Page 608
8.Projective  Spaces • 592 ......Page 610
9.Projective Quadrics • 596 ......Page 614
10.Affine and Projective Spaces • 598  ......Page 616
Bibliography 601 ......Page 619
Index 604......Page 622
cover......Page 1