Book:C.R.J. Clapham/Introduction to Abstract Algebra

Subject Matter

 * Integral Domains
 * The Integers
 * Rings
 * Fields
 * Vector Spaces
 * Field Extensions

Contents

 * Preface


 * 1. Integral Domains
 * 1. Introduction
 * 2. Operations
 * 3. Definition of an Integral Domain
 * 4. Elementary Properties
 * 5. Further Examples of Integral Domains
 * 6. The Residue Classes


 * 2. Ordered and Well-Ordered Integral Domains
 * 7. Order
 * 8. Well-Order


 * 3. The Integers
 * 9. The Principles of Induction
 * 10. Divisibility
 * 11. Highest Common Factor
 * 12. Primes
 * 13. Unique Factorisation


 * 4. Fields
 * 14. Definition of a Field
 * 15. Examples of Fields
 * 16. Subfields
 * 17. The Characteristic of a Field


 * 5. Rings
 * 18. Definition of a Ring
 * 19. Subrings
 * 20. Cosets
 * 21. Ideals
 * 22. Quotient Rings
 * 23. Maximal Ideals
 * 24. Homomorphisms


 * 6. Polynomials and Euclidean Rings
 * 25. Polynomials
 * 26. Divisibility
 * 27. Euclidean Rings
 * 28. Highest Common Factor
 * 29. Irreducible Elements
 * 30. Unique Factorisation
 * 31. Polynomials with Integer Coefficients


 * 7. Vector Spaces
 * 32. Definition of a Vector Space
 * 33. Definition of a Basis
 * 34. Dimension
 * 35. Coordinates


 * 8. Field Extensions
 * 36. The Degree of a Field Extension
 * 37. Roots of a Polynomial
 * 38. Simple Algebraic Extensions
 * 39. A More Sophisticated Approach
 * 40. Constructions with Ruler and Compasses


 * Exercises


 * Answers to the Exercises


 * Index