Algebra Groups Rings & Fields (Hardcover)

Part I: Groups 1. Monoids and Groups 2. How to Divide: Lagrange’s Theorem, Cosets, and an Application to Number Theory 3. Cauchy’s Theorem: How to Show a Number is Greater than 1 4. Introduction to the Classification of Groups: Homomorphisms, Isomorphisms, and Invariants 5. Normal Subgroups? the Building Blocks of the Structure Theory 6. Classifying Groups? Cyclic Groups and Direct Products 7. Finite Abelian Groups 8. Generators and Relations 9. When is a Group a Group? (Cayley’s Theorem) 10. Recounting: Conjugacy Classes and the Class Formula 11. Sylow Subgroups: A New Invariant 12. Solvable Groups: What Could Be Simpler? Part II: Rings and Polynomials 14. An Introduction to Rings 15. The Structure Theory of Rings 16. The Field of Fractions? a Study in Generalization 17. Principal Ideal Domains: Induction without Numbers 18. Roots of Polynomials 19. (Optional) Applications: Famous Results from Number Theory 20. Irreducible Polynomials Part III: Fields 21. Field Extensions: Creating Roots of Polynomials 22. The Problems of Antiquity 23. Adjoining Roots to Polynomials: Splitting 24. Finite Fields 25. The Galois Correspondence 26. Applications of the Galois Correspondence 27. Solving Equations by Radicals