Chapter Ⅰ．Preliminaries-Groups and Rings 1
1．Introduction to Groups 2
2．Quotient Groups and Sylow Subgroups 16
3．Finite Abelian Groups and Solvable Groups 34
4．Introduction to Rings 43
5．Factoring in F[x] 60
Chapter Ⅱ．Field Extensions 72
1．Simple Extensions 72
2．Algebraic Extensions 88
3．Splitting Fields and Normal Extensions 97
Chapter Ⅲ．The Galois Correspondence 110
1．The Fundamental Correspondence 110
2．The Solvable Correspondence 150
Chapter Ⅳ．Applications 163
1．Constructibility 163
2．Roots of Unity 173
3．Wedderburn's Theorem 181
3．Dirichlet's Theorem and Finite Abelian Groups 183
Appendix A-Groups 188
1．Group Actions and the Sylow Theorems 188
2．Free Groups,Generators and Relations 201
Appendix B-Factoring in Integral Domains 211
1．Euclidean Domains and Principal Ideal Domains 211
2．Prime and Irreducible Elements 220
3．Unique Factorization Domains 224
Appendix C-Vector Spaces 230
1．Subspaces,Linear Independence and Spanning 230
2．Bases and Dimension 232
Bibliography 237
Index 239