Book:Iain T. Adamson/Introduction to Field Theory

Subject Matter

 * Field Theory
 * Galois Theory

Contents

 * PREFACE


 * CHAPTER I: ELEMENTARY DEFINITIONS
 * 1. Rings and fields
 * 2. Elementary properties
 * 3. Homomorphisms
 * 4. Vector spaces
 * 5. Polynomials
 * 6. Higher polynomial rings; rational functions
 * Examples I


 * CHAPTER II: EXTENSIONS OF FIELDS
 * 7. Elementary properties
 * 8. Simple extensions
 * 9. Algebraic extensions
 * 10. Factorisation of polynomials
 * 11. Splitting fields
 * 12. Algebraically closed fields
 * 13. Separable extensions
 * Examples II


 * CHAPTER III: GALOIS THEORY
 * 14. Automorphisms of fields
 * 15. Normal extensions
 * 16. The fundamental theorem of Galois theory
 * 17. Norms and traces
 * 18. The primitive element theorem; Lagrange's theorem
 * 19. Normal bases
 * Examples III


 * CHAPTER IV: APPLICATIONS
 * 20. Finite fields
 * 21. Cyclotomic extensions
 * 22. Cyclotomic extensions of the rational number field
 * 23. Cyclic extensions
 * 24. Wedderburn's theorem
 * 25. Ruler-and-compasses constructions
 * 26. Solution by radicals
 * 27. Generic polynomials
 * Examples IV


 * READING LIST
 * INDEX OF NOTATIONS
 * INDEX