Undergraduate Algebra

【文件属性】：
文件名称：Undergraduate Algebra
文件大小：12.45MB
文件格式：PDF
更新时间：2012-06-15 15:40:38
Algebra CHAPTER I The Integers 1 §1. Terminology of Sets 1 §2. Basic Properties 2 §3. Greatest Common Divisor 5 §4. Unique Factorization 7 §5. Equivalence Relations and Congruences 12 CHAPTER II Groups 16 §1. Groups and Examples 16 §2. Mappings 26 §3. Homomorphisms 33 §4. Cosets and Normal Subgroups 41 §5. Application to Cyclic Groups 55 §6. Permutation Groups 59 §7. Finite Abelian Groups 67 §8. Operation of a Group on a Set 73 §9. Sylow Subgroups 79 CHAPTER III Rings 83 §1. Rings 83 §2. Ideals 87 §3. Homomorphisms 90 §4. Quotient Fields 100 X CONTENTS CHAPTER IV Polynomials 105 §1. Polynomials and Polynomial Functions 105 §2. Greatest Common Divisor 118 §3. Unique Factorization 120 §4. Partial Fractions 129 §5. Polynomials Over Rings and Over the Integers 136 §6. Principal Rings and Factorial Rings 143 §7. Polynomials in Several Variables 152 §8. Symmetric Polynomials 159 §9. The Mason-Stothers Theorem 165 §10. The abc Conjecture 171 CHAPTER V Vector Spaces and Modules 177 §1. Vector Spaces and Bases 177 §2. Dimension of a Vector Space 185 §3. Matrices and Linear Maps 188 §4. Modules 192 §5. Factor Modules 203 §6. Free Abelian Groups 205 §7. Modules over Principal Rings 210 §8. Eigenvectors and Eigenvalues 214 §9. Polynomials of Matrices and Linear Maps 220 CHAPTER VI Some Linear Groups 232 §1. The General Linear Group 232 §2. Structure of Gh^(F) 236 §3. SL,(F) 239 §4. SL,(R) and SL,(C) Iwasawa Decompositions 245 §5. Other Decompositions 252 §6. The Conjugation Action 254 CHAPTER VII Field Theory 258 §1. Algebraic Extensions 258 §2. Embeddings 267 §3. Splitting Fields 275 §4. Galois Theory 280 §5. Quadratic and Cubic Extensions 292 §6. Solvability by Radicals 296 §7. Infinite Extensions 302 CHAPTER VIM Finite Fields 309 §1. General Structure 309 §2. The Frobenius Automorphism 313 CONTENTS XI §3. The Primitive Elements 315 §4. Splitting Field and Algebraic Closure 316 §5. Irreducibility of the Cyclotomic Polynomials Over Q 317 §6. Where Does It All Go? Or Rather, Where Does Some of It Go? ... . 321 CHAPTER IX The Real and Complex Numbers 326 §1. Ordering of Rings 326 §2. Preliminaries 330 §3. Construction of the Real Numbers 333 §4. Decimal Expansions 343 §5. The Complex Numbers 346 CHAPTER X Sets 351 §1. More Terminology 351 §2. Zorn's Lemma 354 §3. Cardinal Numbers 359 §4. Well-ordering 369 Appendix §1. The Natural Numbers 373 §2. The Integers 378 §3. Infinite Sets 379 Index 381