Book:Iain T. Adamson/Introduction to Field Theory

Subject Matter

 * Field Theory
 * Galois Theory

Contents



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


 * $\text {II}$: 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 $\text {II}$


 * $\text {III}$: 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 $\text {III}$


 * $\text {IV}$: 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 $\text {IV}$





Source work progress
* : $\S 1.4$: Theorem $4.2$


 * Redoing from start -- need to add the example following this


 * : Chapter $\text {I}$: Elementary Definitions: $\S 4$. Vector Spaces


 * Major refactoring exercise going on in Module Theory and Vector Spaces, which affects the flow of this, so doing that first