Chapter 1 The Integers Page 1 1 1.1 Terminology Sets 2 1.2 Basic Properties 5 1.3 Greatest Common Divisor 7 1.4 Unique Factorization 12 1.5 Equivalence Relations and Congruences Chapter 2 Groups Page 16 16 2.1 Groups and Examples 26 2.2 Mappings 33 2.3 Homomorphisms 41 2.4 Cosets and Normal Subgroups 55 2.5 Application to Cyclic Groups 59 2.6 Permutation Groups 67 2.7 Finite Abelian Groups 73 2.8 Operation of a Group on a Sets 79 2.9 Sylow Subgroups Chapter 3 Rings 83 3.1 Rings 87 3.2 Ideals 90 3.3 Homomorphisms 100 3.4 Quotient Fields Chapter 4 Polynomials Page 105 105 4.1 Polynomials and Polynomial Functions 118 4.2 Greatest Common Divisor 120 4.3 Unique Factorization 129 4.4 Partial Fractions 136 4.5 Polynomials Over Rings and Over the Integers 143 4.6 Principal Rings and Factorial Rings 152 4.7 Polynomials in Several Variables 159 4.8 Symmetric Polynomials 165 4.9 The Mason-Stothers Theorem 171 4.10 The abc Conjecture Chapter 5 Vector Spaces and Modules Page 177 177 5.1 Vector Spaces and Bases 185 5.2 Dimension of a Vector Space 188 5.3 Matrices and Linear Maps 192 5.4 Modules 203 5.5 Factor Modules 205 5.6 Free Abelian Groups 210 5.7 Modules over Principal Rings 214 5.8 Eigenvectors and Eigenvalues 220 5.9 Polynomials of Matrices and Linear Maps Chapter 6 Some Linear Groups Page 232 232 6.1 The General Linear Group 236 6.2 Structure of GL,(F) 239 6.3 SL2(F) 245 6.4 SLn(R) and SL,(C) Iwasawa Decompositions 252 6.5 Other Decompositions 254 6.6 The Conjugation Action Chapter 7 Field Theory Page 258 258 7.1 Algebraic Extensions 267 7.2 Embeddings 275 7.3 Splitting Fields 280 7.4 Galois Theory 292 7.5 Quadratic and Cubic Extensions 296 7.6 Solvability by Radicals 302 7.7 Infinite Extensions Chapter 8 Finite Fields Page 309 309 8.1 General Structure 313 8.2 The Frobenius Automorphism 315 8.3 The Primitive Elements 316 8.4 Splitting Field and Algebraic Closure 317 8.5 Irreducibility of the Cyclotomic Polynomials Over Q 321 8.6 Where Does It All Go? Or Rather, Where Does Some of It Go? Chapter 9 The Real and Complex Numbers Page 326 326 9.1 Ordering of Rings 330 9.2 Preliminaries 333 9.3 Construction of the Real Numbers 343 9.4 Decimal Expansions 346 9.5 The Complex Numbers Chapter 10 Sets Page 353 353 10.1 More Terminology 354 10.2 Zorn's Lemma 359 10.3 Cardinal Numbers 369 10.4 Well-ordering Appendix Page 373 373 1. The Natural Numbers 378 2. The Integers 379 3. Infinite Sets 381 Index