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
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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
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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
