Saunders Mac Lane, Garret Birkhoff9780821816462, 0821816462
This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach–emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s–was popularized for the graduate level in the 1940s and 1950s to some degree by the authors’ publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.
Table of contents :
Contents……Page 11
List of Symbols……Page 17
1. Sets……Page 21
2. Functions……Page 24
3. Relations and Binary Operations……Page 30
4. The Natural Numbers……Page 35
5. Addition and Multiplication……Page 38
6. Inequalities……Page 40
7. The Integers……Page 43
8. The Integers Modulo n……Page 48
9. Equivalence Relations and Quotient Sets……Page 53
10. Morphisms……Page 57
11. Semigroups and Monoids……Page 59
1. Groups and Symmetry……Page 63
2. Rules of Calculation……Page 67
3. Cyclic Groups……Page 71
4. Subgroups……Page 76
5. Defining Relations……Page 79
6. Symmetric and Alternating Groups……Page 83
7. Transformation Groups……Page 88
8. Cosets……Page 92
9. Kernel and Image……Page 95
10. Quotient Groups……Page 99
1. Axioms for Rings……Page 105
2. Constructions for Rings……Page 110
3. Quotient Rings……Page 115
4. Integral Domains and Fields……Page 119
5. The Field of Quotients……Page 121
6. Polynomials……Page 124
7. Polynomials as Functions……Page 129
8. The Division Algorithm……Page 131
9. Principal Ideal Domains……Page 135
10. Unique Factorization……Page 136
11. Prime Fields……Page 140
12. The Euclidean Algorithm……Page 142
13. Commutative Quotient Rings……Page 144
1. Examples of Universals……Page 149
2. Functors……Page 151
3. Universal Elements……Page 154
4. Polynomials in Several Variables……Page 157
5. Categories……Page 161
6. Posets and Lattices……Page 163
7. Contravariance and Duality……Page 166
8. The Category of Sets……Page 173
9. The Category of Finite Sets……Page 176
1. Sample Modules……Page 180
2. Linear Transformations……Page 183
3. Submodules……Page 187
4. Quotient Modules……Page 191
5. Free Modules……Page 193
6. Biproducts……Page 198
7. Dual Modules……Page 205
VI Vector Spaces……Page 213
1. Bases and Coordinates……Page 214
2. Dimension……Page 219
3. Constructions for Bases……Page 222
4. Dually Paired Vector Spaces……Page 227
5. Elementary Operations……Page 232
6. Systems of Linear Equations……Page 239
VII Matrices……Page 243
1. Matrices and Free Modules……Page 244
2. Matrices and Biproducts……Page 252
3. The Matrix of a Map……Page 256
4. The Matrix of a Composite……Page 260
5. Ranks of Matrices……Page 264
6. Invertible Matrices……Page 266
7. Change of Bases……Page 271
8. Eigenvectors and Eigenvalues……Page 277
1. Ordered Domains……Page 281
2. The Ordered Field Q……Page 285
3. Polynomial Equations……Page 287
4. Convergence in Ordered Fields……Page 289
5. The Real Field R……Page 291
6. Polynomials over R……Page 294
7. The Complex Plane……Page 296
8. The Quaternions……Page 301
9. Extended Formal Power Series……Page 304
10. Valuations and p-adic Numbers……Page 306
1. Multilinear and Alternating Functions……Page 313
2. Determinants of Matrices……Page 316
3. Cofactors and Cramer’s Rule……Page 321
4. Determinants of Maps……Page 325
5. The Characteristic Polynomial……Page 329
6. The Minimal Polynomial……Page 332
7. Universal Bilinear Functions ……Page 338
8. Tensor Products ……Page 339
9. Exact Sequences……Page 346
10. Identities on Tensor Products……Page 349
11. Change of Rings……Page 351
12. Algebras……Page 354
1. Bilinear Forms……Page 358
2. Symmetric Matrices……Page 361
3. Quadratic Forms……Page 363
4. Real Quadratic Forms……Page 367
5. Inner Products……Page 371
6. Orthonormal Bases……Page 375
7. Orthogonal Matrices……Page 380
8. The Principal Axis Theorem……Page 384
9. Unitary Spaces……Page 389
10. Normal Matrices……Page 394
1. Noetherian Modules……Page 398
2. Cyclic Modules……Page 401
3. Torsion Modules……Page 403
4. The Rational Canonical Form for Matrices……Page 408
5. Primary Modules……Page 412
6. Free Modules……Page 417
7. Equivalence of Matrices……Page 420
8. The Calculation of Invariant Factors……Page 424
1. Isomorphism Theorems……Page 429
2. Group Extensions……Page 433
3. Characteristic Subgroups……Page 437
4. Conjugate Classes……Page 439
5. The Sylow Theorems……Page 442
6. Nilpotent Groups……Page 446
7. Solvable Groups……Page 448
8. The Jordan-Hölder Theorem……Page 450
9. Simplicity of A_n……Page 453
1. Quadratic and Cubic Equations……Page 456
2. Algebraic and Transcendental Elements……Page 459
3. Degrees……Page 462
4. Ruler and Compass……Page 465
5. Splitting Fields……Page 466
6. Galois Groups of Polynomials……Page 470
7. Separable Polynomials……Page 473
8. Finite Fields……Page 476
9. Normal Extensions……Page 478
10. The Fundamental Theorem……Page 482
11. The Solution of Equations by Radicals……Page 485
1. Posets: Duality Principle……Page 490
2. Lattice Identities……Page 493
3. Sublattices and Products of Lattices……Page 496
4. Modular Lattices……Page 498
5. Jordan-Holder-Dedekind Theorem……Page 500
6. Distributive Lattices……Page 503
7. Rings of Sets……Page 505
8. Boolean Algebras……Page 507
9. Free Boolean Algebras……Page 511
1. Categories……Page 515
2. Functors……Page 521
3. Contravariant Functors……Page 524
4. Natural Transformations……Page 526
5. Representable Functors and Universal Elements……Page 531
6. Adjoint Functors……Page 537
1. Iterated Tensor Products……Page 542
2. Spaces of Tensors……Page 544
3. Graded Modules……Page 550
4. Graded Algebras……Page 553
5. The Graded Tensor Algebra……Page 559
6. The Exterior Algebra of a Module……Page 563
7. Determinants by Exterior Algebra……Page 567
8. Subspaces by Exterior Algebra……Page 572
9. Duality in Exterior Algebra……Page 575
10. Alternating Forms and Skew-Symmetric Tensors……Page 578
1. The Affine Line……Page 581
2. Affine Spaces……Page 584
3. The Affine Group……Page 590
4. Affine Subspaces……Page 596
5. Biaffine and Quadratic Functionals……Page 598
6. Euclidean Spaces……Page 606
7. Euclidean Quadrics……Page 610
8. Projective Spaces……Page 612
9. Projective Quadrics……Page 616
10. Affine and Projective Spaces……Page 618
Bibliography……Page 621
Index……Page 624
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