Real Algebra: A First Course

Description

This book provides an introduction to fundamental methods and techniques of algebra over ordered fields. It is a revised and updated translation of the classic textbook Einführung in die reelle Algebra. Beginning with the basics of ordered fields and their real closures, the book proceeds to discuss methods for counting the number of real roots of polynomials. Followed by a thorough introduction to Krull valuations, this culminates in Artin's solution of Hilbert's 17th Problem. Next, the fundamental concept of the real spectrum of a commutative ring is introduced with applications. The final chapter gives a brief overview of important developments in real algebra and geometry―as far as they are directly related to the contents of the earlier chapters―since the publication of the original German edition. Real Algebra is aimed at advanced undergraduate and beginning graduate students who have a good grounding in linear algebra, field theory and ring theory. It also provides a carefully written reference for specialists in real algebra, real algebraic geometry and related fields. Preface Preface to Einführung in die reelle Algebra Contents 1 Ordered Fields and Their Real Closures 1.1 Orderings and Preorderings of Fields 1.2 Quadratic Forms, Witt Rings, Signatures 1.3 Extension of Orderings 1.4 The Prime Ideals of the Witt Ring 1.5 Real Closed Fields and Their Field Theoretic Characterization 1.6 Galois Theoretic Characterization of Real Closed Fields 1.7 Counting Real Zeroes of Polynomials (without Multiplicities) 1.8 Conceptual Interpretation of the Sylvester Form 1.9 Cauchy Index of a Rational Function, Bézoutian and Hankel Forms 1.10 An Upper Bound for the Number of Real Zeroes (with Multiplicities) 1.11 The Real Closure of an Ordered Field 1.12 Transfer of Quadratic Forms 2 Convex Valuation Rings and Real Places 2.1 Convex Subrings of Ordered Fields 2.2 Valuation Rings 2.3 Integral Elements 2.4 Valuations, Ideals of Valuation Rings 2.5 Residue Fields and Subfields of Convex Valuation Rings 2.6 The Topology of Ordered and Valued Fields 2.7 The Baer–Krull Theorem 2.8 Places 2.9 The Orderings of R(t), ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper R left parenthesis left parenthesis t right parenthesis right parenthesis) /StPNE pdfmark [/StBMC pdfmarkR((t))ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper Q u o t double struck upper R left brace t right brace) /StPNE pdfmark [/StBMC pdfmark`3́9`42`"̇613A``45`47`"603AQuotR{t}ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 2.10 Composition and Decomposition of Places 2.11 Existence of Real Places of Function Fields 2.12 Artin's Solution of Hilbert's 17th Problem and the Sign Change Criterion 3 The Real Spectrum 3.1 The Zariski Spectrum. Affine Varieties 3.2 Reality for Commutative Rings 3.3 Definition of the Real Spectrum 3.4 Constructible Subsets and Spectral Spaces 3.5 The Geometric Setting: Semialgebraic Sets and Filter Theorems 3.6 The Space of Closed Points 3.7 Specializations and Convex Ideals 3.8 The Real Spectrum and the Reduced Witt Ring of a Field 3.9 Preorderings of Rings and Positivstellensätze 3.10 The Convex Radical Ideals Associated to a Preordering 3.11 Boundedness 3.12 Prüfer Rings and the Real Holomorphy Ring of a Field 4 Recent Developments 4.1 Counting Real Solutions 4.2 Quadratic Forms 4.3 Stellensätze 4.4 Noncommutative Stellensätze References Symbol Index Index