Author | Ribenboim, Paulo. author |
---|---|
Title | The Theory of Classical Valuations [electronic resource] / by Paulo Ribenboim |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0551-7 |
Descript | XI, 403 p. online resource |
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