Chapter 1 Number Theory 1
Section 1.1 Induction 1
Section 1.2 Binomial Theorem and Complex Numbers 18
Section 1.3 Greatest Common Divisors 37
Section 1.4 The Fundamental Theorem of Arithmetic 55
Section 1.5 Congruences 59
Section 1.6 Dates and Days 76
Chapter 2 Groups Ⅰ 84
Section 2.1 Some Set Theory 84
Functions 87
Equivalence Relations 99
Section 2.2 Permutations 106
Section 2.3 Groups 125
Symmetry 137
Section 2.4 Subgroups and Lagrange's Theorem 147
Section 2.5 Homomorphisms 159
Section 2.6 Quotient Groups 171
Section 2.7 Group Actions 192
Section 2.8 Counting with Groups 208
Chapter 3 Commutative Rings Ⅰ 217
Section 3.1 First Properties 217
Section 3.2 Fields 230
Section 3.3 Polynomials 235
Section 3.4 Homomorphisms 243
Section 3.5 From Numbers to Polynomials 252
Euclidean Rings 267
Section 3.6 Unique Factorization 275
Section 3.7 Irreducibility 281
Section 3.8 Quotient Rings and Finite Fields 290
Section 3.9 A Mathematical Odyssey 305
Latin Squares 305
Magic Squares 310
Design of Experiments 314
Proiective Planes 316
Chapter 4 Linear Algebra 320
Section 4.1 Vector Spaces 320
Gaussian Elimination 344
Section 4.2 Euclidean Constructions 354
Section 4.3 Linear Transformations 366
Section 4.4 Eigenvalues 383
Section 4.5 Codes 399
Block Codes 399
Linear Codes 406
Decoding 423
Chapter 5 Fields 432
Section 5.1 Classical Formulas 432
Viète's Cubic Formula 444
Section 5.2 Insolvability of the General Quintic 449
Formulas and Solvability by Radicals 459
Quadratics 460
Cubics 461
Quartics 461
Translation into Group Theory 462
Section 5.3 Epilog 471
Chapter 6 Groups Ⅱ 475
Section 6.1 Finite Abelian Groups 475
Section 6.2 The Sylow Theorems 489
Section 6.3 Ornamental Symmetry 501
Chapter 7 Commutative Rings Ⅱ 518
Section 7.1 Prime Ideals and Maximal Ideals 518
Section 7.2 Unique Factorization 525
Section 7.3 Noetherian Rings 535
Section 7.4 Varieties 540
Section 7.5 Generalized Divison Algorithm 558
Monomial Orders 559
Division Algorithm 565
Section 7.6 Gr？bner Bases 570