代数学

Chapter Ⅰ Set theory and Number Theory

1 Set Theory

2 Unique Factorization Theorem

3 Congruence

4 Chinese Remainder Theorem

5 Complex Integers

6 Real Numbers and p-aclic Numbers

Chapter Ⅱ Group theory

1 Definitions

2 The Transformation Groups on Sets

3 Subgroups

4 Normal Subgroups and Inner Automorphisms

5 Automorphism Groups

6 p-Groups and Sylow Theorems

7 Jordan-Holder Theorem

8 Symmetric Group Sn

Chapter Ⅲ Polynomials

1 Fields and Rings

2 Polynomial Rings and Quotient Fields

3 Unique Factorization Theorem for Polynomials

4 Symmetric Polynomial, Resultant and Discriminant

5 Ideals

Chapter Ⅳ Linear Algebra

1 Vector Spaces

2 Basis and Dimension

3 Linear Transformation and Matrix

4 Module and Module over P.I.D

5 Jordan Canonical Form

6 Characteristic Polynomial

7 Inner Product and Bilinear form

8 Spectral Theory

Chapter Ⅴ Polynomials in One Variable and Field Theory

1 Algebraically Closed Field

2 Algebraic Extension

3 Algebraic Closure

4 Characteristic and Finite Field

5 Separable Algebraic Extension

6 Galois Theory

7 Solve Equation by Radicals

8 Field Polynomial and Field Discriminant

9 Luroth's Theorem

Appendix

A1 Set Theoretical Notations

A2 Peano's Axioms

A3 Homological Algebra

Index