LEADER 03927nam 2200493 450 001 996495168503316 005 20231110233816.0 010 $a3-031-09800-5 035 $a(MiAaPQ)EBC7120735 035 $a(Au-PeEL)EBL7120735 035 $a(CKB)25188964500041 035 $a(PPN)265856426 035 $a(EXLCZ)9925188964500041 100 $a20230310d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aReal algebra $ea first course /$fManfred Knebusch, Claus Scheiderer 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (217 pages) 225 1 $aUniversitext 311 08$aPrint version: Knebusch, Manfred Real Algebra Cham : Springer International Publishing AG,c2022 9783031097997 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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`3?9`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. 327 $a3.3 Definition of the Real Spectrum -- 3.4 Constructible Subsets and Spectral Spaces -- 3.5 The Geometric Setting: Semialgebraic Sets and Filter Theorems -- 3.6 The Space of Closed Points -- 3.7 Specializations and Convex Ideals -- 3.8 The Real Spectrum and the Reduced Witt Ring of a Field -- 3.9 Preorderings of Rings and Positivstellensätze -- 3.10 The Convex Radical Ideals Associated to a Preordering -- 3.11 Boundedness -- 3.12 Prüfer Rings and the Real Holomorphy Ring of a Field -- 4 Recent Developments -- 4.1 Counting Real Solutions -- 4.2 Quadratic Forms -- 4.3 Stellensätze -- 4.4 Noncommutative Stellensätze -- References -- Symbol Index -- Index. 410 0$aUniversitext 606 $aAlgebra 606 $aÀlgebra$2thub 608 $aLlibres electrònics$2thub 615 0$aAlgebra. 615 7$aÀlgebra 676 $a512 700 $aKnebusch$b Manfred$054845 702 $aScheiderer$b Claus 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996495168503316 996 $aReal algebra$93058447 997 $aUNISA