Chapter 1 Number Theory

1.1．Induction

1.2．Binomial Coefficients

1.3．Greatest Common Divisors

1.4．The Fundamental Theorem of Arithmetic

1.5．Congruences

1.6．Dates and Days

Chapter 2 Groups Ⅰ

2.1．Functions

2.2．Permutations

2.3．Groups

Symmetry

2.4．Lagrange's Theorem

2.5．Homomorphisms

2.6．Quotient Groups

2.7．Group Actions

2.8．Counting with Groups

Chapter 3 Commutative Rings Ⅰ

3.1．First Properties

3.2．Fields

3.3．Polynomials

3.4．Homomorphisms

3.5．Greatest Common Divisors

Euclidean Rings

3.6．Unique Factorization

3.7．Irreducibility

3.8．Quotient Rings and Finite Fields

3.9．Officers,Fertilizer,and a Line at Infinity

Chapter 4 Goodies

4.1．Linear Algebra

Vector Spaces

Linear Transformations

Applications to Fields

4.2．Euclidean Constructions

4.3．Classical Formulas

4.4．Insolvability of the General Quintic

Formulas and Solvability by Radicals

Translation into Group Theory

4.5．Epilog

Chapter 5 Groups Ⅱ

5.1．Finite Abelian Groups

5.2．The Sylow Theorems

5.3．The Jordan-H？1der Theorem

5.4．Presentations

Chapter 6 Commutative Rings Ⅱ

6.1．Prime Ideals and Maximal Ideals

6.2．Unique Factorization

6.3．Noetherian Rings

6.4．Varieties

6.5．Gr？bner Bases

Generalized Division Algorithm

Gr？bner Bases

Hints to Exercises

Bibliography

Index