1 Groups, Rings and Fields

Chapters in this book

1. Frontmatter I
2. Preface V
3. Contents VII
4. 1 Groups, Rings and Fields 1
5. 2 Maximal and Prime Ideals 21
6. 3 Prime Elements and Unique Factorization Domains 29
7. 4 Polynomials and Polynomial Rings 53
8. 5 Field Extensions 66
9. 6 Field Extensions and Compass and Straightedge Constructions 80
10. 7 Kronecker’s Theorem and Algebraic Closures 91
11. 8 Splitting Fields and Normal Extensions 113
12. 9 Groups, Subgroups and Examples 119
13. 10 Normal Subgroups, Factor Groups and Direct Products 141
14. 11 Symmetric and Alternating Groups 161
15. 12 Solvable Groups 171
16. 13 Groups Actions and the Sylow Theorems 180
17. 14 Free Groups and Group Presentations 192
18. 15 Finite Galois Extensions 217
19. 16 Separable Field Extensions 233
20. 17 Applications of Galois Theory 248
21. 18 The Theory of Modules 265
22. 19 Finitely Generated Abelian Groups 285
23. 20 Integral and Transcendental Extensions 295
24. 21 The Hilbert Basis Theorem and the Nullstellensatz 312
25. 22 Algebraic Cryptography 326
26. Bibliography 359
27. Index 363