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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 Adeles, Ideles, 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 Class 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

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 Adeles, Ideles, 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 Class 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

Library of Congress subject headings for this publication: Algebraic number theory