域上二次型引论
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作者T.Y.,Lam著,T.,Y.,LAM 译
出版社高等教育出版社
ISBN9787040469196
出版时间2018-06
版次1
装帧精装
开本16开
纸张胶版纸
定价199元
上书时间2024-05-21
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基本信息
书名:域上二次型引论
定价：199.00元
作者:T.Y.,Lam著,T.,Y.,LAM 译
出版社：高等教育出版社
出版日期：2018-06-01
ISBN：9787040469196
字数：
页码：
版次：1
装帧：精装
开本：16开
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内容提要
作为作者获奖书Algebraic Theory of Quadratic Forms (Benjamin, 1973) 的新版，本书给出了在特征非 2 的任意域上的二次型理论的一个现代、自足的导引。从除了线性代数外的少量预备知识出发，作者讲述了一个专家级的课程，内容从二次型的Witt经典理论、四元数与Clifford 代数、形式实域的 Artin-Schreier理论、Witt 环的结构定理，到 Pfister形式理论、函数域和域不变量。这些主要进展与所涉及的 Brauer-Wall 群、局部与整体域、迹形式、Galois理论以及初等代数 K-理论天衣无缝地交织在一起，对域上二次型理论做了一个原创性处理。新版中增加了超过100页全新的两章，内容包括这个领域中更新的结果以及更加近代的观点。作为作者写作的特点，本书主要内容的陈述总是穿插大量精心挑选的解释一般理论的例题。这个特点再加上全部十三章280多个内容丰富的习题，极大提升了本书的价值，使得本书可以作为代数、数论、代数几何、代数拓扑以及几何拓扑研究者的参考书。
目录
PrefaceNotes to the ReaderPartial List of NotationsChapter I. Foundations1. Quadratic Forms and Quadratic Spaces2. Diagonalizatioof Quadratic Forms3. Hyperbolic Plane and Hyperbolic Spaces4. DecompositioTheorem and CancellatioTheorem5. Witt's ChaiEquivalence Theorem6. Kronecker Product of Quadratic Spaces7. Generatioof the Orthogonal Group by ReflectionsExercises for Chapter IChapter II. Introductioto Witt Rings1. Definitioof W(F) and W(F)2. Group of Square Classes3. Some Elementary Computations4. Presentatioof Witt Rings5. Classificatioof Small Witt RingsExercises for Chapter IIChapter III. QuaternioAlgebras and their Norm Forms1. Constructioof QuaternioAlgebras2. QuaternioAlgebras as Quadratic Spaces3. Coverings of the Orthogonal Groups4. Linkage of QuaternioAlgebras5. Characterizations of QuaternioAlgebras Exercises for Chapter IIIChapter IV. The Brauer-Wall Group1. The Brauer Group2. Central Simple Graded Algebras (CSGA)3. Structure Theory of CSGA4. The Brauer-Wall Group Exercises for Chapter IVChapter V. Clifford Algebras1. Constructioof Clifford Algebras2. Structure Theorems3. The Clifford Invariant, Witt Invariant, and Hasse Invariant4. Real Periodicity and Clifford Modules5. Compositioof Quadratic Forms6. Steinberg Symbols and Milnor's Group k2FExercises for Chapter VChapter VI. Local Fields and Global Fields1. Springer's Theorem for C.D.V. Fields2. Quadratic Forms over Local FieldsAppendix: Nonreal Fields with Four Square Classes3. Hasse-Minkowski Principle4. Witt Ring of Q5. Hilbert Reciprocity and Quadratic ReciprocityExercises for Chapter VIChapter VII. Quadratic Forms Under Algebraic Extensions1. Scharlau's Transfer2. Simple Extensions and Springer's Theorem3. Quadratic Extensions4. Scharlau's Norm Principle5. Knebusch's Norm Principle6. Galois Extensions and Trace Forms7. Quadratic Closures of FieldsExercises for Chapter VIIChapter VIII. Formally Real Fields, Real-Closed Fields, and PythagoreaFields1. Structure of Formally Real Fields2. Characterizations of Real-Closed FieldsAppendix A: Uniqueness of Real-ClosureAppendix B: Another Artin-Schreier Theorem3. Pfister's Local-Global Principle4. PythagoreaFieldsAppendix: Fields with 8 Square Classes and 20rderings5. Connections with Galois Theory6. HarrisoTopology oXF7. Prime Spectrum of W(F)8. Applications to the Structure of W(F)9. AIntroductioto PreorderingsExercises for Chapter VIIIChapter IX. Quadratic Forms under Transcendental Extensions1. Cassels-Pfister Theorem2. Second and Third RepresentatioTheorems3. Milnor's Exact Sequence for W(F(x))4. Scharlau's Reciprocity Formula for F(x)Exercises for Chapter IXChapter X. Pfister Forms and FunctioFields1. ChaiP-EquivalenceAppendix: Round Forms2. Multiplicative Forms3. Introductioto FunctioFields4. Basic Theorems oFunctioFields5. Hanptsatz, Linkage, and Forms iInF6. Milnor's Higher K-Groups Exercises for Chapter XChapter XI. Field Invariants1. Sums of Squares2. The Level of a Field3. Pfister-Witt Annihilator Theorem4. The Property (An)5. Height and Pythagoras Number6. The u-Invariant of a FieldAppendix: The General u-Invariant7. The Size of W(F), and C-FieldsExercises for Chapter XIChapter XII. Special Topics iQuadratic Forms1. Isomorphisms of Witt Rings2. Quadratic Forms of Low DimensionAppendix: Forms with Isomorphic FunctioFields3. Some ClassificatioTheorems4. Witt Rings under Biquadratic Extensions5. Nonreal Fields with Eight Square Classes6. Kaplansky Radical and Hilbert Fields7. Constructioof Some Pre-Hilbert Fields8. Axiomatic Schemes for Quadratic FormsExercises for Chapter XIIChapter XIII. Special Topics oInvariants1. The u-Invariant of C((x, y))2. Fields of u-Invariant 63. Fields of Pythagoras Number 6 and 74. Levels of Commutative Rings5. Pythagoras Numbers of Commutative Rings6. Some OpeQuestionsExercises for Chapter XIIIBibliographyIndex
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