Introductory Notes on Valuation Rings and Function Fields in One Variable

1. Front Matter

   Pages i-viii

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2. Introduction Generalities on algebraic functions of one variabile

   • Renata Scognamillo, Umberto Zannier

   Pages 1-6

3. Basic notions on function fields of one variable

   • Renata Scognamillo, Umberto Zannier

   Pages 7-29

4. Valuation rings

   • Renata Scognamillo, Umberto Zannier

   Pages 31-84

5. Back Matter

   Pages 85-119

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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.

• Dedekind domains
• Hilbert's Nullstellensatz
• Puiseux series
• valuation rings

• Scuola Normale Superiore, Pisa, Italy

  Renata Scognamillo, Umberto Zannier

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