The Theory of Classical Valuations / Edition 1

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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.

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Editorial Reviews

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"It is well written, encyclopedic, and authoritative and probably belongs on the shelf of any commutative algebraist or algebraic number theorist."—MATHEMATICAL REVIEWS

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Product Details

  • ISBN-13: 9780387985251
  • Publisher: Springer New York
  • Publication date: 5/21/1999
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 1999
  • Edition number: 1
  • Pages: 403
  • Product dimensions: 6.14 (w) x 9.21 (h) x 1.00 (d)

Table of Contents

Introduction 1
Ch. 1 Absolute Values of Fields 3
Ch. 2 Valuations of a Field 55
Ch. 3 Polynomials and Henselian Valued Fields 79
Ch. 4 Extensions of Valuations 107
Ch. 5 Uniqueness of Extensions of Valuations and Poly-Complete Fields 127
Ch. 6 Extensions of Valuations: Numerical Relations 151
Ch. 7 Power Series and the Structure of Complete Valued Fields 185
Ch. 8 Decomposition and Inertia Theory 213
Ch. 9 Ramification Theory 227
Ch. 10 Valuation Characterizations of Dedekind Domains 263
Ch. 11 Galois Groups of Algebraic Extensions of Infinite Degree 307
Ch. 12 Ideals, Valuations, and Divisors in Algebraic Extensions of Infinite Degree over Q 333
Ch. 13 A Glimpse of Krull Valuations 365
App Commutative Fields and Characters of Finite Abelian Groups 373
Bibliography 391
Index 395
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