Table of Contents

1 Groups, Rings and Fields

2 Maximal and Prime Ideals

3 Prime Elements and Unique Factorization Domains

4 Polynomials and Polynomial Rings

5 Field Extensions

6 Field Extensions and Compass and Straightedge Constructions

7 Kronecker's Theorem and Algebraic Closures

8 Splitting Fields and Normal Extensions

9 Groups, Subgroups and Examples

10 Normal Subgroups, Factor Groups and Direct Products

11 Symmetric and Alternating Groups

12 Solvable Groups

13 Groups Actions and the Sylow Theorems

14 Free Groups and Group Presentations

15 Finite Galois Extensions

16 Separable Field Extensions

17 Applications of Galois Theory

18 The Theory of Modules

19 Finitely Generated Abelian Groups

20 Integral and Transcendental Extensions

21 The Hilbert Basis Theorem and the Nullstellensatz

22 Algebraic Cryptography

About the Author

Celine ?Carstensen, Volkswohlbund Insurance, Dortmund, Germany;Benjamin Fine, Fairfield University, Connecticut, USA; Gerhard Rosenberger, Universitat Hamburg, Germany.