Chapter 1 Number Theory 1
1.1．Induction 1
1.2．Binomial Coefficients 17
1.3．Greatest Common Divisors 36
1.4．The Fundamental Theorem of Arithmetic 58
1.5．Congruences 62
1.6．Dates and Days 73
Chapter 2 Groups Ⅰ 82
2.1．Functions 82
2.2．Permutations 97
2.3．Groups 115
Symmetry 128
2.4．Lagrange's Theorem 134
2.5．Homomorphisms 143
2.6．Quotient Groups 156
2.7．Group Actions 178
2.8．Counting with Groups 194
Chapter 3 Commutative Rings Ⅰ 203
3.1．First Properties 203
3.2．Fields 216
3.3．Polynomials 225
3.4．Homomorphisms 233
3.5．Greatest Common Divisors 239
Euclidean Rings 252
3.6．Unique Factorization 261
3.7．Irreducibility 267
3.8．Quotient Rings and Finite Fields 278
3.9．Officers,Fertilizer,and a Line at Infinity 289
Chapter 4 Goodies 301
4.1．Linear Algebra 301
Vector Spaces 301
Linear Transformations 318
Applications to Fields 329
4.2．Euclidean Constructions 332
4.3．Classical Formulas 345
4.4．Insolvability of the General Quintic 363
Formulas and Solvability by Radicals 368
Translation into Group Theory 371
4.5．Epilog 381
Chapter 5 Groups Ⅱ 385
5.1．Finite Abelian Groups 385
5.2．The Sylow Theorems 397
5.3．The Jordan-H？1der Theorem 408
5.4．Presentations 420
Chapter 6 Commutative Rings Ⅱ 437
6.1．Prime Ideals and Maximal Ideals 437
6.2．Unique Factorization 445
6.3．Noetherian Rings 456
6.4．Varieties 462
6.5．Gr？bner Bases 480
Generalized Division Algorithm 482
Gr？bner Bases 493
Hints to Exercises 505
Bibliography 519
Index 521