Algebraic Number Theory

Title page
Preface
References
Chapter 1 Elementary Valuation Theory
1-1. Valuations and Prime Divisors
1-2. The Approximation Theorem
1-3. Archimedean and Nonarchimedean Prime Divisors
1-4. The Prime Divisors of Q
1-5. Fields with a Discrete Prime Divisor
1-6. e and f
1-7. Completions
1-8. The Theorem of Ostrowski
1-9. Complete Fields with Discrete Prime Divisor
Exercises
Chapter 2 Extension of Valuations
2-1. Uniqueness of Extensions (Complete Case)
2-2. Existence of Extensions (Complete Case)
2-3. Extensions of Discrete Prime Divisors
2-4. Extensions in the General Case
2-5. Consequences
Exercises
Chapter 3 Local Fields
3-1. Newton's Method
3-2. Unramified Extensions
3-3. Totally Ramified Extensions
3-4. Tamely Ramified Extensions
3-5. Inertia Group
3-6. Ramification Groups
3-7. Different and Discriminant
Exercises
Chapter 4 Ordinary Arithmetic Fields
4-1. Axioms and Basic Properties
4-2. Ideals and Divisors
4-3. The Fundamental Theorem of OAFs
4-4. Dedekind Rings
4-5. Over-rings of O
4-6. Class Number
4-7. Mappings of Ideals
4-8. Different and Discriminant
4-9. Factoring Prime Ideals in an Extension Field
4-10. Hilbert Theory
Exercises
Chapter 5 Global Fields
5-1. Global Fields and the Product Formula
5-2. Adèles, Idèles, Divisors, and Ideals
5-3. Unit Theorem and Class Number
5-4. Class Number of an Algebraic Number Field
5-5. Topological Considerations
5-6. Relative Theory
Exercises
Chapter 6 Quadratic Fields
6-1. Integral Basis and Discriminant
6-2. Prime Ideals
6-3. Units
6-4. C1ass Number
6-5. The Local Situation
6-6. Norm Residue Symbol
Chapter 7 Cyclotomic Fields
7-1. Elementary Facts
7-2. Unramified Primes
7-3. Quadratic Reciprocity Law
7-4. Ramified Primes
7-5. Integral Basis and Discriminant
7-6. Units
7-7. Class Number
Symbols and Notation
Index