The theory of classical valuations

  title={The theory of classical valuations},
  author={Paulo Ribenboim},
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… 

A family of totally ordered groups with some special properties

Let K be a field with a Krull valuation | | and value group G ¬= {1}, and let B K be the valuation ring. Theories about spaces of countable type and Hilbert-like spaces in [1] and spaces of

Summary on non-Archimedean valued fields

This article summarizes the main properties of ultrametric spaces, valued fields, ordered fields and fields with valuations of higher rank, highlighting their similarities and differences. The most

Another Proof of the Existence a Dedekind Complete Totally Ordered Field

We describe the Dedekind cuts explicitly in terms of non-standard rational num­ bers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another

GanzstellensäTze in Theories of Valued Fields

This paper uses model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of functions over a valued field which are integral definite on some definable set.

Model Theory of Valued fields

These notes focus mainly on the model theory of algebraically closed valued fields (loosely referred to as ACVF). This subject begins with work by A. Robinson in the 1950s (see the proof of model

Integral closure of ideals, rings, and modules

Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7.


Simple valuation ideals of order 3 in two-dimensional regular local rings

Let (R, m) be a 2-dimensional regular local ring with alge- braically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is