The Theory of Classical Valuations

1. Front Matter

   Pages i-xi

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2. Introduction

   • Paulo Ribenboim

   Pages 1-1

3. Absolute Values of Fields

   • Paulo Ribenboim

   Pages 3-54

4. Valuations of a Field

   • Paulo Ribenboim

   Pages 55-78

5. Polynomials and Henselian Valued Fields

   • Paulo Ribenboim

   Pages 79-105

6. Extensions of Valuations

   • Paulo Ribenboim

   Pages 107-125

7. Uniqueness of Extensions of Valuations and Poly-Complete Fields

   • Paulo Ribenboim

   Pages 127-150

8. Extensions of Valuations: Numerical Relations

   • Paulo Ribenboim

   Pages 151-184

9. Power Series and the Structure of Complete Valued Fields

   • Paulo Ribenboim

   Pages 185-212

10. Decomposition and Inertia Theory

    • Paulo Ribenboim

    Pages 213-226

11. Ramification Theory

    • Paulo Ribenboim

    Pages 227-261

12. Valuation Characterizations of Dedekind Domains

    • Paulo Ribenboim

    Pages 263-306

13. Galois Groups of Algebraic Extensions of Infinite Degree

    • Paulo Ribenboim

    Pages 307-331

14. Ideals, Valuations, and Divisors in Algebraic Extensions of Infinite Degree over ℚ

    • Paulo Ribenboim

    Pages 333-364

15. A Glimpse of Krull Valuations

    • Paulo Ribenboim

    Pages 365-372

16. Back Matter

    Pages 373-403

    PDF

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.

• Algebra
• Division
• Finite
• Topology
• commutative property
• function
• fundamental theorem
• theorem

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

• Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

  Paulo Ribenboim

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