Abstract Algebra: Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography

Description

A new approach to conveying abstract algebra, the area that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras, that is essential to various scientific disciplines such as particle physics and cryptology. It provides a well written account of the theoretical foundations and it also includes a chapter on cryptography. End of chapter problems help readers with accessing the subjects. Covering topics not found in competing works: Free groups, module theory, extensions of rings New chapter about Groups, Algebras and Representations New section about the hot topic Combinatorial and geometric group theory Preface Preface to the second edition 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. Algebras and group representations 23. Algebraic cryptography Bibliography Index