The present book comes from the first part of the lecture notes I used for a first-year graduate algebra course at the University of Minnesota, Purdue University, and Peking University. The Chinese versions of these notes were published by The Peking University Press in 1986, and by Linking Publishing Co of Taiwan in 1987.
The aim of this book is not only to give the student quick access to the basic knowledge of algebra, either for future advancement in the field of algebra, or for general background information, but also to show that algebra is truly a master key or a 'skeleton key' to many mathematical problems. As one knows, the teeth of an ordinary key prevent it from opening all but one door, whereas the skeleton key keeps only the essential parts, allowing it to unlock many doors.
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
