Algebra

Table of Contents
......Page 4
1. Basic Operations......Page 16
2. Row Reduction......Page 24
3. Determinants......Page 33
4. Permutation Matrices......Page 39
5. Cramer's Rule......Page 43
Exercises
......Page 46
1. Definition of a Group
......Page 53
2. Subgroups
......Page 59
3. Isomorphisms
......Page 63
4. Homomorphisms
......Page 66
5. Equivalence Relations and Partitions
......Page 68
6. Cosets
......Page 72
7. Restriction of a Homomorphism to a Subgroup
......Page 74
8. Products of Groups
......Page 76
9. Modular Arithmetic
......Page 79
10. Quotient Groups
......Page 81
Exercises
......Page 84
1. Real Vector Spaces
......Page 93
2. Abstract Fields
......Page 97
3. Bases and Dimension
......Page 102
4. Computation with Bases
......Page 109
5. Infinite Dimensional Spaces
......Page 115
6. Direct Sums
......Page 117
Exercises
......Page 119
1. The Dimension Formula
......Page 124
2. The Matrix of a Linear Transformation
......Page 126
3. Linear Operators and Eigenvectors
......Page 130
4. The Characteristic Polynomial
......Page 135
5. Orthogonal Matrices and Rotations
......Page 138
6. Diagonalization
......Page 145
7. Systems of Differential Equations
......Page 148
8. The Matrix Exponential
......Page 153
Exercises
......Page 160
1. Symmetry of Plane Figures......Page 170
2. Group of Motions of the Plane......Page 172
3. Finite Groups of Motions......Page 177
4. Discrete Groups of Motions......Page 181
5. Abstract Symmetry: Group Operations......Page 190
6. The Operation on Cosets......Page 193
7. The Counting Formula......Page 195
8. Permutation Representations......Page 197
9. Finite Subgroups of the Rotation Group......Page 199
Exercises
......Page 203
1. Operations of a Group on Itself
......Page 212
2. Class Equation of the Icosahedral Group
......Page 215
3. Operations on Subsets
......Page 218
4. The Sylow Theorems
......Page 220
5. The Groups of Order 12
......Page 224
6. Computation in the Symmetric Group
......Page 226
7. The Free Group
......Page 232
8. Generators and Relations
......Page 234
9. The Todd-Coxeter Algorithm
......Page 238
Exercises
......Page 244
1. Definition of Bilinear Form......Page 252
2. Symmetric Forms: Orthogonality
......Page 258
3. The Geometry Associated to a Positive Form
......Page 262
4. Hermitian Forms
......Page 264
5. The Spectral Theorem
......Page 268
6. Conics and Quadrics
......Page 270
7. The Spectral Theorem for Normal Operators
......Page 274
8. Skew Symmetric Forms
......Page 275
9. Summary of Results, in Matrix Notation
......Page 276
Exercises
......Page 277
1. The Classical Linear Groups
......Page 285
2. The Special Unitary Group SU2
......Page 287
3. The Orthogonal Representation of SU2
......Page 291
4. The Special Linear Group SL2(R)
......Page 296
5. One-Parameter Subgroups
......Page 298
6. The Lie Algebra
......Page 301
7. Translation in a Group
......Page 307
8. Simple Groups
......Page 310
Exercises
......Page 315
1. Definition of a Group Representation
......Page 322
2. G-Invariant Forms and Unitary Representations
......Page 325
3. Compact Groups
......Page 327
4. G-Invariant Subspaces and Irreducible Representations
......Page 329
5. Characters
......Page 331
6. Permutation Representations and Regular Representation
......Page 336
7. The Representations of the Icosahedral Group
......Page 338
9. Schur's Lemma, and Proof of the Orthogonality Relations
......Page 340
10. Representations of the Group SU2
......Page 345
Exercises
......Page 350
1. Definition of a Ring
......Page 360
2. Formal Construction of Integers and Polynomials
......Page 362
3. Homomorphisms and Ideals
......Page 368
4. Quotient Rings and Relations in a Ring
......Page 374
5. Adjunction of Elements
......Page 379
6. Integral Domains and Fraction Fields
......Page 383
7. Maximal Ideals
......Page 385
8. Algebraic Geometry
......Page 388
Exercises
......Page 394
1. Factorization of Integers and Polynomials
......Page 404
2. Unique Factorization Domains, Pricipal Ideal Domains, and Euclidean Domains
......Page 407
3. Gauss's Lemma
......Page 413
4. Explicit Factorization of Polynomials
......Page 417
5. Primes in the Ring of Gauss Integers
......Page 421
6. Algebraic Integers
......Page 424
7. Factorization in Imaginary Quadratic Fields
......Page 429
8. Ideal Factorization
......Page 434
9. The Relation Between Prime Ideals of R and Prime Integers
......Page 439
10. Ideal Classes in Imaginary Quadratic Fields
......Page 440
11. Real Quadratic Fields
......Page 448
12. Some Diophantine Equations
......Page 452
Exercises
......Page 455
1. Definition of a Module
......Page 465
2. Matrices, Free Modules, and Bases
......Page 467
3. The Principle of Permanence of Identities
......Page 471
4. Diagonalization of Integer Matrices
......Page 472
5. Generators and Relations for Modules
......Page 479
6. The Structure Theorem for Abelian Groups
......Page 486
7. Application to Linear Operators
......Page 491
8. Free Modules Over Polynomial Rings
......Page 497
Exercises
......Page 498
1. Examples of Fields
......Page 507
2. Algebraic and Transcendental Elements
......Page 508
3. The Degree of a Field Extension
......Page 511
4. Constructions with Ruler and Compass......Page 515
5. Symbolic Adjunction of Roots
......Page 521
6. Finite Fields
......Page 524
7. Function Fields
......Page 530
8. Transcendental Extensions
......Page 540
9. Algebraically Closed Fields
......Page 542
Exercises
......Page 545
1. The Main Theorem of Galois Algebra
......Page 552
2. Cubic Equations
......Page 558
3. Symmetric Functions
......Page 562
4. Primitive Elements
......Page 567
5. Proof of the Main Theorem
......Page 571
6. Quartic Equations
......Page 575
7. Kummer Extensions
......Page 580
8. Cyclotomic Extensions
......Page 582
9. Quintic Equations
......Page 585
Exercises
......Page 590
1. Set Theory
......Page 600
2. Techniques of Proof
......Page 604
3. Topology
......Page 608
4. The Implicit Function Theorem
......Page 612
Exercises
......Page 614
Suggestions for Further Reading
......Page 618
Index
......Page 622