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