The Theory of Classical Valuations

1 Absolute Values of Fields.- 1.1. First Examples.- 1.2. Generalities About Absolute Values of a Field.- 1.3. Absolute Values of Q.- 1.4. The Independence of Absolute Values.- 1.5. The Topology of Valued Fields.- 1.6. Archimedean Absolute Values.- 1.7. Topological Characterizations of Valued Fields.- 2 Valuations of a Field.- 2.1. Generalities About Valuations of a Field.- 2.2. Complete Valued Fields and Qp.- 3 Polynomials and Henselian Valued Fields.- 3.1. Polynomials over Valued Fields.- 3.2. Henselian Valued Fields.- 4 Extensions of Valuations.- 4.1. Existence of Extensions and General Results.- 4.2. The Set of Extensions of a Valuation.- 5 Uniqueness of Extensions of Valuations and Poly-Complete Fields.- 5.1. Uniqueness of Extensions.- 5.2. Poly-Complete Fields.- 6 Extensions of Valuations: Numerical Relations.- 6.1. Numerical Relations for Valuations with Unique Extension.- 6.2. Numerical Relations in the General Case.- 6.3. Some Interesting Examples.- 6.4. Appendix on p-Groups.- 7 Power Series and the Structure of Complete Valued Fields.- 7.1. Power Series.- 7.2. Structure of Complete Discrete Valued Fields.- 8 Decomposition and Inertia Theory.- 8.1. Decomposition Theory.- 8.2. Inertia Theory.- 9 Ramification Theory.- 9.1. Lower Ramification Theory.- 9.2. Higher Ramification.- 10 Valuation Characterizations of Dedekind Domains.- 10.1. Valuation Properties of the Rings of Algebraic Integers.- 10.2. Characterizations of Dedekind Domains.- 10.3. Characterizations of Valuation Domains.- 11 Galois Groups of Algebraic Extensions of Infinite Degree.- 11.1. Galois Extensions of Infinite Degree.- 11.2. The Abelian Closure of Q.- 12 Ideals, Valuations, and Divisors in Algebraic Extensions of Infinite Degree over Q.- 12.1. Ideals.- 12.2. Valuations, Dedekind Domains, and Examples.- 12.3. Divisors of Algebraic Number Fields of Infinite Degree.- 13 A Glimpse of Krull Valuations.- 13.1. Generalities.- 13.2. Integrally Closed Domains.- 13.3. Suggestions for Further Study.- Appendix Commutative Fields and Characters of Finite Abelian Groups.- A.1. Algebraic Elements.- A.2. Algebraic Elements, Algebraically Closed Fields.- A.3. Algebraic Number Fields.- A.4. Characteristic and Prime Fields.- A.5. Normal Extensions and Splitting Fields.- A.6. Separable Extensions.- A.7. Galois Extensions.- A.8. Roots of Unity.- A.9. Finite Fields.- A.10. Trace and Norm of Elements.- A.11. The Discriminant.- A.12. Discriminant and Resultant of Polynomials.- A.13. Inseparable Extensions.- A.14. Perfect Fields.- A.15. The Theorem of Steinitz.- A.16. Orderable Fields.- A.17. The Theorem of Artin.- A.18. Characters of Finite Abelian Groups.