The Theory of Classical Valuations [Paperback]

Valuation theory is used constantly in algebraic number theory and field theory, and is currently gaining considerable research interest. Ribenboim fills a unique niche in the literature as he presents one of the first introductions to classical valuation theory in this up-to-date rendering of the authors long-standing experience with the applications of the theory. The presentation is fully up-to-date and will serve as a valuable resource for students and mathematicians.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.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 Fiel“+