Algebra abstract and modern

Cover......Page 1
Contents......Page 6
Preface......Page 10
Part I: Preliminaries......Page 12
  1.1 Sets and subsets......Page 14
  1.2 Relations and functions......Page 22
  1.3 Equivalence relations and partitions......Page 32
  1.4 The cardinality of a set......Page 38
  2.1 Integers......Page 48
  2.2 Congruence modulo n......Page 60
  2.3 Rational, real and complex numbers......Page 70
  2.4 Ordering......Page 77
  2.5 Matrices......Page 81
  2.6 Determinants......Page 90
Part II: Group Theory......Page 104
  3.1 Binary systems......Page 106
  3.2 Groups......Page 119
  3.3 Elementary properties of groups......Page 135
  3.4 Finite groups and group tables......Page 148
 Chapter 4: Subgroups and Quotient Groups......Page 156
  4.1 Subgroups......Page 157
  4.2 Cyclic groups......Page 167
  4.3 Cosets of a subgroup......Page 179
  4.4 Lagrange’s theorem......Page 185
  4.5 Normal subgroups......Page 194
  4.6 Quotient groups......Page 200
 Chapter 5: Homomorphisms of Groups......Page 208
  5.1 Definition and examples......Page 209
  5.2 Fundamental theorem of homomorphisms......Page 223
  5.3 Isomorphism theorems......Page 230
  5.4 Automorphisms......Page 236
  6.1 Cayley’s theorem......Page 246
  6.2 The symmetric group Sn......Page 252
  6.3 Cycles......Page 256
  6.4 Alternating group An and dihedral group Dn......Page 268
  7.1 Action of a group on a set......Page 286
  7.2 Orbits and stabilizers......Page 293
  7.3 Certain counting techniques......Page 304
  7.4 Cauchy and Sylow theorems......Page 313
  8.1 Direct products......Page 326
  8.2 Finitely generated abelian groups......Page 337
  8.3 Invariants of finite abelian groups......Page 354
  8.4 Groups of small order......Page 358
Part III: Ring Theory......Page 362
  9.1 Examples and elementary properties......Page 364
  9.2 Certain special elements in rings......Page 377
  9.3 The characteristic of a ring......Page 383
  9.4 Subrings......Page 386
  9.5 Homomorphisms of rings......Page 390
  9.6 Certain special types of rings......Page 396
  9.7 Integral domains and fields......Page 404
  10.1 Ideals......Page 414
  10.2 Quotient rings......Page 433
  10.3 Chinese remainder theorem......Page 442
  10.4 Prime ideals......Page 447
  10.5 Maximal ideals......Page 456
  10.6 Embeddings of rings......Page 469
  11.1 Rings of polynomials......Page 482
  11.2 The division algorithm......Page 496
  11.3 Polynomials over a field......Page 506
  11.4 Irreducible polynomials......Page 512
 Chapter 12: Factorization in Integral Domains......Page 518
  12.1 Divisibility in integral domains......Page 519
  12.2 Principal ideal domains......Page 527
  12.3 Unique factorization domains......Page 535
  12.4 Polynomials over UFDs......Page 543
  12.5 Euclidean domains......Page 553
  12.6 Some applications to number theory......Page 561
 Chapter 13: Modules and Vector Spaces......Page 568
  13.1 Modules and submodules......Page 569
  13.2 Homomorphisms and quotients of modules......Page 577
  13.3 Direct products and sums......Page 585
  13.4 Simple and completely reducible modules......Page 598
  13.5 Free modules......Page 602
  13.6 Vector spaces......Page 609
Part IV: Field Theory......Page 620
  14.1 Extensions of a field......Page 622
  14.2 Algebraic extensions......Page 627
  14.3 Algebraically closed fields......Page 639
  14.4 Derivatives and multiple roots......Page 646
  14.5 Finite fields......Page 652
  15.1 Separable and normal extensions......Page 658
  15.2 Automorphism groups and fixed fields......Page 667
  15.3 Fundamental theorem of Galois theory......Page 676
 Chapter 16: Selected Applications of Galois Theory......Page 684
  16.1 Fundamental theorem of algebra......Page 685
  16.2 Cyclic extensions......Page 688
  16.3 Solvable groups......Page 691
  16.4 Polynomials solvable by radicals......Page 694
  16.5 Constructions by ruler and compass......Page 703
Answers/Hints to Selected Even-Numbered Exercises......Page 710
Index......Page 774