Algebra : groups, rings, and fields
著者
書誌事項
Algebra : groups, rings, and fields
A K Peters, c1994
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注記
Includes index
内容説明・目次
内容説明
This text presents the concepts of higher algebra in a comprehensive and modern way for self-study and as a basis for a high-level undergraduate course. The author is one of the preeminent researchers in this field and brings the reader up to the recent frontiers of research including never-before-published material. From the table of contents: - Groups: Monoids and Groups - Cauchyis Theorem - Normal Subgroups - Classifying Groups - Finite Abelian Groups - Generators and Relations - When Is a Group a Group? (Cayley's Theorem) - Sylow Subgroups - Solvable Groups - Rings and Polynomials: An Introduction to Rings - The Structure Theory of Rings - The Field of Fractions - Polynomials and Euclidean Domains - Principal Ideal Domains - Famous Results from Number Theory - I Fields: Field Extensions - Finite Fields - The Galois Correspondence - Applications of the Galois Correspondence - Solving Equations by Radicals - Transcendental Numbers: e and p - Skew Field Theory - Each chapter includes a set of exercises
目次
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
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