文件名称：Advanced Algebra
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更新时间：2012-06-15 15:02:15
Algebra
CONTENTS ContentsofBasic Algebra x Preface xi List of Figures xv Dependence amongChapters xvi Guide forthe Reader xvii Notation and Terminology xxi I. TRANSITIONTO MODERNNUMBERTHEORY 1 1. Historical Background 1 2. Quadratic Reciprocity 8 3. Equivalence and Reduction of Quadratic Forms 12 4. Composition of Forms, ClassGroup 24 5. Genera 31 6. Quadratic Number Fieldsand Their Units 35 7. Relationship of Quadratic Formsto Ideals 38 8. Primesin the Progressions4n +1 and 4n +350 9. Dirichlet Seriesand Euler Products 56 10. Dirichlet’sTheorem onPrimesin Arithmetic Progressions 61 11. Problems 67 II. WEDDERBURN–ARTINRING THEORY 76 1. Historical Motivation 77 2. Semisimple Ringsand Wedderburn’sTheorem 81 3. Ringswith Chain Condition and Artin’sTheorem 87 4. Wedderburn–Artin Radical 89 5. Wedderburn’sMain Theorem 94 6. Semisimplicity and Tensor Products 104 7. Skolem–Noether Theorem 111 8. Double Centralizer Theorem 114 9. Wedderburn’sTheorem about Finite Division Rings 117 10. Frobenius’sTheorem about Division Algebrasover the Reals 118 11. Problems 120 vii viii Contents III. BRAUERGROUP 123 1. Definition and Examples, Relative Brauer Group 124 2. Factor Sets 132 3. CrossedProducts 135 4. Hilbert’sTheorem 90 145 5. Digression on Cohomology ofGroups 147 6. Relative Brauer Groupwhen the GaloisGroupIsCyclic 158 7. Problems 162 IV. HOMOLOGICAL ALGEBRA 166 1. Overview 167 2. Complexesand Additive Functors 171 3. Long Exact Sequences 184 4. Projectivesand Injectives 192 5. Derived Functors 202 6. Long Exact SequencesofDerived Functors 210 7. Ext and Tor 223 8. Abelian Categories 232 9. Problems 250 V. THREE THEOREMSINALGEBRAICNUMBERTHEORY 262 1. Setting 262 2. Discriminant 266 3. Dedekind Discriminant Theorem 274 4. Cubic Number FieldsasExamples 279 5. Dirichlet Unit Theorem 288 6. Finitenessof the ClassNumber 298 7. Problems 307 VI. REINTERPRETATIONWITH ADELESANDIDELES 313 1. p-adic Numbers 314 2. Discrete Valuations 320 3. Absolute Values 331 4. Completions 342 5. Hensel’sLemma 349 6. Ramification Indicesand Residue ClassDegrees 353 7. Special Featuresof GaloisExtensions 368 8. Different and Discriminant 371 9. Global and Local Fields 382 10. Adelesand Ideles 388 11. Problems 397 Contents ix VII. INFINITEFIELDEXTENSIONS 403 1. Nullstellensatz 404 2. Transcendence Degree 408 3. Separable and Purely Inseparable Extensions 414 4. Krull Dimension 423 5. Nonsingular and Singular Points 428 6. Infinite GaloisGroups 434 7. Problems 445 VIII.BACKGROUNDFORALGEBRAICGEOMETRY 447 1. Historical Originsand Overview 448 2. Resultant and Bezout’sTheorem 451 3. Projective Plane Curves 456 4. Intersection Multiplicity for a Line with a Curve 466 5. Intersection Multiplicity for Two Curves 473 6. General Form of Bezout’sTheorem for Plane Curves 488 7. Gr¨obner Bases 491 8. Constructive Existence 499 9. Uniquenessof Reduced Gr¨obner Bases 508 10. SimultaneousSystemsofPolynomial Equations 510 11. Problems 516 IX. THE NUMBERTHEORYOF ALGEBRAICCURVES 520 1. Historical Originsand Overview 520 2. Divisors 531 3. Genus 534 4. Riemann–Roch Theorem 540 5. Applications of the Riemann–Roch Theorem 552 6. Problems 554 X. METHODSOF ALGEBRAICGEOMETRY 558 1. Affine Algebraic Setsand Affine Varieties 559 2. Geometric Dimension 563 3. Projective Algebraic Setsand Projective Varieties 570 4. Rational Functionsand Regular Functions 579 5. Morphisms 590 6. Rational Maps 595 7. Zariski’sTheorem about Nonsingular Points 600 8. Classification Questionsabout Irreducible Curves 604 9. Affine Algebraic SetsforMonomial Ideals 618 10. Hilbert Polynomial in the Affine Case 626 x Contents X. METHODSOF ALGEBRAICGEOMETRY (Continued) 11. Hilbert Polynomial in the Projective Case 633 12. Intersectionsin Projective Space 635 13. Schemes 638 14. Problems 644 HintsforSolutionsof Problems 649 Selected References 713 Index of Notation 717 Index 721 CONTENTS OF BASIC ALGEBRA I. Preliminaries about the Integers, Polynomials, and Matrices II. Vector SpacesoverQ,R,andC III. Inner-Product Spaces IV. Groupsand GroupActions V. Theory of a Single Linear Transformation VI. Multilinear Algebra VII. Advanced GroupTheory VIII. Commutative Ringsand Their Modules IX. Fieldsand GaloisTheory X. Modulesover Noncommutative Rings