CHAPTER 1. Algebraic Extensions |
1. Definitions |
2. Algebraic extensions |
3. Characteristic: perfect fields |
4. Separability of extensions |
5. Normal extensions |
6. Finite fields |
7. Primitive elements |
8. Algebraically closed fields |
9. Norms and traces |
EXERCISES |
CHAPTER 2. Galois Theory |
1. Automorphisms of extensions: Galois extensions |
2. The fundamental theorem of Galois theory |
3. An example |
4. Cyclotomic fields |
5. The first cohomolgy group |
6. Cyclic extensions |
7. Multiplicative Kummer theory |
8. Additive Kummer theory |
9. Solutions of polynomial equations by radicals |
10. Infinite Galois extensions |
11. The Krull topology |
12. Inverse limits |
EXERCISES |
CHAPTER 3. Introduction to Valuation Theory |
1. Definition of valuation: examples |
2. Valuations on the fields Q and k(x) |
3. Complete fields and completions |
4. Value groups and residue class fields |
5. Prolongations of valuations |
6. Relatively complete fields |
7. "Prolongations of valuations, continued" |
EXERCISES |
CHAPTER 4. Extensions of Valuated Fields |
1. Ramification and residue class degree |
2. Unramified and tamely ramified extensions |
3. The different |
4. Extensions K/k with K/k separable |
5. Ramification groups |
EXERCISES |
CHAPTER 5. Dedekind Fields |
1. The fundamental theorem of Dedekind fields |
2. Extensions of Dedekind fields |
3. Factoring of ideals in extensions |
4. Galois extensions of Dedekind fields |
EXERCISES |
APPENDIX 1. Proof of Theorem 19 of Chapter 2 |
APPENDIX 2. Example of the Galois Group of an Infinite Extension |
BIBLIOGRAPHY |
INDEX |