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```PREFACE                                    vii
PREFACE TO THE SECOND EDITION              ix
TO THE STUDENT                            xiii
WRITING PROOFS                             xvi
HISTORICAL BACKGROUND                      xxi
1 INTEGERS                                  1
1.1  Divisors  ..................  . .... .......  3
1.2  Primes  ............... . .............. .  15
1.3  Congruences  .................... ........  24
1.4  Integers Modulo n  .............. .. . .......  35
N otes  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46
2 FUNCTIONS                                47
2.1  Functions  ...............................  49
2.2  Equivalence Relations ................ . .......... .  62
2.3  Permutations  ..................  .........  71
Notes  ............. ......................  86
3 GROUPS                                   87
3.1  Definition of a Group  .............. . . . . ........  88
3.2  Subgroups ..............................  102
3.3  Constructing Examples  .................  ..... . .  115
3.4  Isomorphisms  ................... ........  124
3.5  Cyclic Groups ................... ........  135
3.6  Permutation Groups ............ . .... ....... ..  142
3.7  Homomorphisms  ...................   ......  152
3.8  Cosets, Normal Subgroups, and Factor Groups . ..........  164
N otes  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  177
4  POLYNOMIALS                                                   179
4.1 Fields; Roots of Polynomials ........    ........ .. . 179
4.2  Factors  ............    .....   ....  . ..  .......  .  192
4.3  Existence of Roots . .................. . ......         203
4.4  Polynomials over Z, Q, R, and C . . . . . . . .  .  .  .  .  .   .  211
Notes  .............. ......           ......... ........     222
5  COMMUTATIVE RINGS                                             223
5.1 Commutative Rings; Integral Domains . . . . . . . . . . . . ...  224
5.2  Ring Homomorphisms . .................. ...          . ..236
5.3 Ideals and Factor Rings ... . . . . . . . .  .   . .  ... .  251
5.4  Quotient Fields  ................      .   . . . .  . ... . .  262
Notes ............. .... .             ...     .........      268
6  FIELDS                                                        269
6.1 Algebraic Elements ................... . ...... .         270
6.2  Finite and Algebraic Extensions . . . . . . . . . . . . . ...  276
6.3  Geometric Constructions ................... . .      . ..283
6.4  Splitting Fields ..  . . . ........    ....  ...... . . . 289
6.5  Finite Fields  .......... . .    . ....   ........   .   295
6.6 Irreducible Polynomials over Finite Fields . . . . . . . . . . ...  301
6.7  Quadratic Reciprocity . . . . . . . . . .  . .  . . . . . . . 307
Notes ....... . ..... .......            ............ .       314
7  STRUCTURE OF GROUPS                                           315
7.1 Isomorphism Theorems; Automorphisms . . . . . . . . . . ...  316
7.2  Conjugacy .......... . . ....... ...........             323
7.3  Groups Acting on Sets .... . . . . . . .    . . . ...... .  330
7.4  The Sylow  Theorems  . ................     .   ..... . .. 338
7.5  Finite Abelian Groups .. . . . . . . . .  .   . . . . .  .  342
7.6  Solvable Groups ...... . . . . . . . . . . . .  ...... .  350
7.7  Simple Groups  . ..................        ........  . .. 357
8  GALOIS THEORY                                                 365
8.1 The Galois Group of a Polynomial . . . . . . . . . . . . ....  366
8.2  Multiplicity of Roots  . . . . . . . . . . . ...... ... ... . .  372
8.3  The Fundamental Theorem of Galois Theory  . . . . . . . . ...  376
8.4  Solvability by Radicals ..... . . . . .  . . .  .   . . . .  .  386
8.5  Cyclotomic Polynomials ..... . . . . . . . .  .   . . .  .  392
8.6  Computing Galois Groups ... . . . . . .  .    .  .   .. . . 397
9  UNIQUE FACTORIZATION                                           407
9.1  Principal Ideal Domains ................ .... . 408
9.2  Unique Factorization Domains .........    ......... . 415
9.3  Some Diophantine Equations ........ . .............      421
APPENDIX                                                         433
A.1 Sets .................            .............         . 433
A.2 Construction of the Number Systems . . . . . . ... . ...... .  436
A.3 Basic Properties of the Integers . . . . . . . . . . . . . . ...  439
A.4 Induction .................. ..          .    .  .  .  .  .  .   .  440
A.5 Complex Numbers . . . . . . . . . . . . . . .  . ...... . 444
A.6 Solution of Cubic and Quartic Equations . . . . . . . . . . ....  450
A.7 Dimension of a Vector Space .. . . . . . . . . .  . . .. . . 458
BIBLIOGRAPHY                                                      461