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Birkhäuser - Birkhäuser Mathematics - Scuola Normale Superiore | Introductory Notes on Valuation Rings and Function Fields in One Variable

Introductory Notes on Valuation Rings and Function Fields in One Variable

Scognamillo, Renata, Zannier, Umberto

2014, VIII, 119 p.

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  • Provides the basic theory of valuation rings and shows applications to many issues treated only individually in other books
  • Presents the theory of valuation rings in an elementary and general way and results that which are difficult to locate in the literature
  • Gives several arithmetical applications of the main theory and other important results from geometry and number theory
The book deals with the (elementary and introductory) theory of valuation rings. As explained in the introduction, this represents a useful and important viewpoint in algebraic geometry, especially concerning the theory of algebraic curves and their function fields. The correspondences of this with other viewpoints (e.g. of geometrical or topological nature) are often indicated, also to provide motivations and intuition for many results. Links with arithmetic are also often indicated. There are three appendices, concerning Hilbert’s Nullstellensatz (for which several proofs are provided), Puiseux series and Dedekind domains. There are also several exercises, often accompanied by hints, which sometimes develop further results not included in full for brevity reasons.

Content Level » Graduate

Keywords » Dedekind domains - Hilbert's Nullstellensatz - Puiseux series - valuation rings

Related subjects » Scuola Normale Superiore

Table of contents 

Generalities on function fields of one variable.- Valuation rings.- Completions.- Appendices on Hilbert's Nullstellensatz, Puiseux series, Dedekind domains.

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