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Full Description
In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.
Contents
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.
