1 | Field Extensions | 6 |
---|---|---|
1.1 Basic Facts | 6 | |
1.2 Basic Examples | 9 | |
1.3 Norm, Trace and Discriminant | 12 | |
2 | Ring Extensions | 15 |
2.1 Basic Processes in Ring Theory | 15 | |
2.2 Noetherian Rings and Modules | 17 | |
2.3 Integral Extensions | 19 | |
2.4 Discriminant of a Number Field | 21 | |
3 | Dedekind Domains and Ramification Theory | 26 |
3.1 Dedekind Domains | 27 | |
3.2 Extensions of Primes | 32 | |
3.3 Kummer's Theorem | 35 | |
3.4 Dedekind's Discriminant Theorem | 37 | |
3.5 Ramification in Galois Extensions | 38 | |
3.6 Decomposition and Inertia Groups | 40 | |
3.7 Quadratic and Cyclotomic Extensions | 42 | |
4 | Class Number and Lattices | 46 |
4.1 Norm of an ideal | 46 | |
4.2 Embeddings and Lattices | 48 | |
4.3 Minkowski's Theorem | 52 | |
4.4 Finiteness of Class Number and Ramification | 53 | |
Bibliography | 56 | |
A | Appendix: Notes on Galois Theory | 57 |
A.1 Preamble | 57 | |
A.2 Field Extensions | 58 | |
A.3 Splitting Fields and Normal Extensions | 60 | |
A.4 Separable Extensions | 62 | |
A.5 Galois Theory | 63 | |
A.6 Norms and Traces | 67 | |
B | Appendix: Discriminants in Algebra and Arithmetic | 70 |
B.1 Discriminant in High School Algebra | 70 | |
B.2 Discriminant in College Algebra | 74 | |
B.3 Discriminant in Arithmetic | 77 | |
References | 82 | |
Gzipped Postscript File | PDF File |