Real algebra : a first course / / Manfred Knebusch, Claus Scheiderer

• Autore: Knebusch Manfred
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (217 pages)
• Disciplina: 512
• Soggetto topico: Algebra; Àlgebra
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): ScheidererClaus
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Real algebra
• ISBN: 3-031-09800-5
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996495168503316
• Lo trovi qui: Univ. di Salerno
• Serie: Universitext