Introductory Notes on Valuation Rings and Function Fields in One Variable

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.

List of contents

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

Summary

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.

Product details

• Authors: Renat Scognamillo, Renata Scognamillo, Umberto Zannier
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 14.07.2015
• EAN: 9788876425004
• ISBN: 978-88-7642-500-4
• No. of pages: 119
• Dimensions: 150 mm x 10 mm x 239 mm
• Weight: 205 g
• Illustrations: VIII, 119 p.
• Series: Publications of the Scuola Normale Superiore / Theses (Scuola Normale Superiore); Publications of the Scuola Normale Superiore; Publications of the Scuola Normale Superiore / Lecture Notes (Scuola Normale Superiore); Lecture Notes (Scuola Normale Superiore); Publications of the Scuola Nor; Publications of the Scuola Normale Superiore; Publications of the Scuola Normale Superiore / Theses (Scuola Normale Superiore); Lecture Notes (Scuola Normale Superiore)
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, B, geometry, Mathematics and Statistics, Number Theory