Synopses & Reviews
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.
Review
"It is well written, encyclopedic, and authoritative and probably belongs on the shelf of any commutative algebraist or algebraic number theorist."--MATHEMATICAL REVIEWS
Synopsis
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.
Table of Contents
Absolute Values of Fields.- Valuations of a Field.- Polynomials and Henselian Valued Fields.- Extensions of Valuations.- Uniqueness of Extensions of Valuations and Poly-Complete Fields.- Extensions of Valuations: Numerical Relations.- Power Series and the Structure of Complete Valued Fields.- Decomposition and Inertia Theory.- Ramification Theory.- Valuation Characterization of Dedekind Domains.- Galois Groups of Algebraic Extensions of Infinite Degree.- Ideals, Valuations and Divisors in Algebraic Extensions of Infinite Degree of the Field of Rational Numbers.- A Glimpse on Krull Valuations.