Contents

Kapitel in diesem Buch

1. Frontmatter I
2. Preface V
3. Preface to the second edition VII
4. Contents IX
5. 1. Groups, rings and fields 1
6. 2. Maximal and prime ideals 21
7. 3. Prime elements and unique factorization domains 29
8. 4. Polynomials and polynomial rings 53
9. 5. Field extensions 67
10. 6. Field extensions and compass and straightedge constructions 81
11. 7. Kronecker’s theorem and algebraic closures 93
12. 8. Splitting fields and normal extensions 119
13. 9. Groups, subgroups, and examples 125
14. 10. Normal subgroups, factor groups, and direct products 147
15. 11. Symmetric and alternating groups 169
16. 12. Solvable groups 179
17. 13. Groups actions and the Sylow theorems 189
18. 14. Free groups and group presentations 201
19. 15. Finite Galois extensions 227
20. 16. Separable field extensions 243
21. 17. Applications of Galois theory 257
22. 18. The theory of modules 275
23. 19. Finitely generated Abelian groups 293
24. 20. Integral and transcendental extensions 303
25. 21. The Hilbert basis theorem and the nullstellensatz 319
26. 22. Algebras and group representations 333
27. 23. Algebraic cryptography 365
28. Bibliography 399
29. Index 403