Contents Introduction . xi 1 Group Theory . 1 1.1 Definitions and Examples . 1 1.2 Subgroups and Group Homomorphisms . 4 1.3 Group Constructions .

8 1.4 The Isomorphism Theorems . 13 1.5 Group Actions . 15 1.6 Cyclic Groups . 16 1.7 Permutation Groups .

20 1.8 p-Groups and the Sylow Theorems . 27 1.9 Solvable and Nilpotent Groups . 30 1.10 Free Groups and Presentations . 35 1.11 Further Topics .

39 1.12 Further Reading . 40 Exercises . 42 2 Commutative Algebra . 45 2.1 Rings . 45 2.2 Ideals .

48 2.3 Polynomials . 52 2.4 Modules . 56 2.5 Module Constructions . 60 2.6 Noetherian Modules .

63 2.7 Prime and Maximal Ideals . 66 2.8 Localization . 70 2.9 Gauss''s Lemma . 76 2.10 Principal Ideal Domains .

78 2.11 Field Extensions . 85 2.12 Finite Fields . 89 2.13 Further Topics . 92 2.14 Further Reading .

93 Exercises . 95 3 Linear Algebra . 99 3.1 Vector Spaces . 99 viii Contents 3.2 Dimension . 103 3.3 Vector Space Constructions .

107 3.4 Eigenvalues and Eigenvectors . 111 3.5 The Determinant . 115 3.6 Matrices . 120 3.7 Matrix Operations .

125 3.8 Inner Product Spaces . 131 3.9 Matrix Decompositions . 138 3.10 Further Topics . 142 3.11 Further Reading .

143 Exercises . 145 4 Topology . 149 4.1 Definitions and Examples . 149 4.2 Continuity . 153 4.3 Topological Space Constructions .

156 4.4 Separation Axioms . 159 4.5 Connectedness . 163 4.6 Compactness . 166 4.7 Tychonoff''s Theorem .

170 4.8 Metric Spaces . 173 4.9 Completeness . 179 4.10 Homotopy . 182 4.11 Further Topics .

186 4.12 Further Reading . 188 Exercises . 189 5 Real Analysis . 193 5.1 Limits . 193 5.2 Infinite Series .

197 5.3 Uniform Convergence . 200 5.4 Differentiation on R . 204 5.5 Taylor''s Theorem . 210 5.6 Measurable Spaces .

214 5.7 Measurable Functions . 217 5.8 Integration . 222 5.9 Measure Extensions . 230 5.10 Borel Measure .

235 5.11 The Fundamental Theorem of Calculus . 238 Contents ix 5.12 Further Topics . 242 5.13 Further Reading . 244 Exercises . 246 6 Multivariable Analysis .

249 6.1 Multivariable Differentiation . 249 6.2 Multivariable Integration . 256 6.3 The Change of Variables Formula . 259 6.4 Differential Equations .

265 6.5 Common Derivatives and Integrals . 268 6.6 The Gaussian Integral . 272 6.7 The Weierstrass Approximation Theorem . 276 6.8 The Constant Rank Theorem .

284 6.9 Further Topics . 290 6.10 Further Reading . 292 Exercises . 293 7 Complex Analysis . 295 7.1 Contour Integrals .

296 7.2 The Jordan Curve Theorem . 302 7.3 The Topology of Contours . 308 7.4 Green''s Theorem . 316 7.5 The Cauchy-Riemann Equations .

321 7.6 Cauchy''s Integral Formula . 324 7.7 Consequences of Cauchy''s Integral Formula . 327 7.8 Meromorphic Functions . 332 7.9 Residues .

338 7.10 The Open Mapping Theorem . 341 7.11 Tauberian Theorems . 345 7.12 Further Topics . 348 7.13 Further Reading .

349 Exercises . 351 8 Number Theory . 353 8.1 Galois Theory . 353 8.2 Algebraic Integers . 359 8.3 Prime Factorization in Ok .

363 8.4 Quadratic Fields . 368 8.5 Cyclotomic Extensions . 371 8.6 Diophantine Equations . 373 8.7 Quadratic Reciprocity .

378 8.8 Solvability by Radicals . 381 8.9 The Riemann ζ-Function . 386 8.10 The Prime Number Theorem . 390 8.11 Further Topics .

394 8.12 Further Reading . 396 Exercises . 398 9 Probability . 401 9.1 Definitions and Constructions . 401 9.2 Densities .

404 9.3 Lp spaces . 408 9.4 The Radon-Nikodym Theorem . 413 9.5 Mean and Variance . 417 9.6 Joint Density Functions .

422 9.7 Common Probability Distributions . 425 9.8 Convergence of Distributions . 432 9.9 Higher Moments and Characteristic Functions . 438 9.10 The Central Limit Theorem .

444 9.11 Further Topics . 445 9.12 Further Reading . 447 Exercises . 448 Appendix . 451 A.1 Set Theory .

451 A.2 The Axiom of Choice . 455 A.3 Cardinality .