Mathematics 129, Spring 2009 Number Fields

1. Concerning These Notes 2 2. Introduction (First Day) 2 3. Algebraic Numbers and Integers 5 4. Linear Algebra for Number Theory 10 5. Review of Field Theory; Traces/Norms 10 6. Units: Some Notes Missing Here 16 7. Diophantine Approximation 16 8. The Trace Pairing 18 9. Linear Algebra and Discriminants 21 10. The Ring of Integers Inside the Number Field 26 11. Some Computational Aspects of Discriminants 28 12. Cyclotomic Fields 33 13. Noetherian Modules and Noetherian Rings 36 14. Ideals 39 15. Dedekind Domains 40 16. Ideals in Dedekind Domains 42 17. Norms of Ideals 47 18. Geometry of Numbers 49 19. Geometry of Canonical Embedding; Class Groups 51 20. Hermite’s Theorem on Discriminants 55 21. Dirichlet’s Unit Theorem 57 22. Rings of Fractions / Localisation 61 23. Splitting of Primes 65 24. Splitting in Quadratic Fields 66 25. Splitting of Primes – the Main Theorems 69 26. The Discriminant and Ramification 73 27. Quadratic Reciprocity 78 28. An example Class Number Formula 81 29. Galois extensions 83 30. Relative Norms of Ideals 86