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TRANSITION TO MODERN NUMBER THEORY 1 1. Historical Background 1 2. Quadratic Reciprocity 8 3. Equivalence and Reduction of Quadratic Forms 12 4. Composition of Forms, Class Group 24 5. Genera 31 6. Quadratic Number Fields and Their Units 35 7. Relationship of Quadratic Forms to Ideals 38 8. Primes in the Progressions 4n + 1 and 4n + 3 50 9. Dirichlet Series and Euler Products 56 10. Dirichlet's Theorem on Primes in Arithmetic Progressions 61 11. Problems 67 I. WEDDERBURN-ARTIN RING THEORY 76 1. Historical Motivation 77 2. Semisimple Rings and Wedderburn's Theorem 81 3. Rings with Chain Condition and Artin's Theorem 87 4. Wedderburn-Artin Radical 89 5. Wedderburn's Main Theorem 1 94 6. Semisimplicity and Tensor Products 104 7. Skolem-Noether Theorem 111 8. Double Centralizer Theorem 114 9. Wedderburn's Theorem about Finite Division Rings 117 10. Frobenius's Theorem about Division Algebras over the Reals 118 11. Problems 120 III. BRAUER GROUP 123 1. Definition and Examples, Relative Brauer Group 124 2. Factor Sets 132 3. Crossed Products 135 4. Hilbert's Theorem 90 145 5. Digression on Cohomology of Groups 147 6. Relative Brauer Group when the Galois Group Is Cyclic 158 7. Problems 162 IV. HOMOLOGICAL ALGEBRA 166 1. Overview 167 2. Complexes and Additive Functors 171 3. Long Exact Sequences 184 4. Projectives and Injectives 192 5. Derived Functors 202 6. Long Exact Sequences of Derived Functors 210 7. Ext and Tor 223 8. Abelian Categories 232 9. Problems 250 V. THREE THEOREMS IN ALGEBRAIC NUMBER THEORY 262 1. Setting 262 2. Discriminant 266 3. Dedekind Discriminant Theorem 274 4. Cubic Number Fields as Examples 279 5. Dirichlet Unit Theorem 288 6. Finiteness of the Class Number 298 7. Problems 307 VI. REINTERPRETATION WITH ADELES AND IDELES 313 1. p-adic Numbers 314 2. Discrete Valuations 320 3. Absolute Values 331 4. Completions 342 5. Hensel's Lemma 349 6. Ramification Indices and Residue Class Degrees 353 7. Special Features of Galois Extensions 368 8. Different and Discriminant 371 9. Global and Local Fields 382 10. Adeles and Ideles 388 11. Problems 397 VII. INFINITE FIELD EXTENSIONS 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 Galois Groups 434 7. Problems 445 VIII. BACKGROUND FOR ALGEBRAIC GEOMETRY 447 1. Historical Origins and Overview 448 2. Resultant and Bezout's Theorem 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's Theorem for Plane Curves 488 7. Gribner Bases 491 8. Constructive Existence 499 9. Uniqueness of Reduced Gribner Bases 508 10. Simultaneous Systems of Polynomial Equations 510 11. Problems 516 IX. THE NUMBER THEORY OF ALGEBRAIC CURVES 520 1. Historical Origins and 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. METHODS OF ALGEBRAIC GEOMETRY 558 1. Affine Algebraic Sets and Affine Varieties 559 2. Geometric Dimension 563 3. Projective Algebraic Sets and Projective Varieties 570 4. Rational Functions and Regular ,Functions 579 5. Morphisms 590 6. Rational Maps 595 7. Zariski's Theorem about Nonsingular Points 600 8. Classification Questions about Irreducible Curves 604 9. Affine Algebraic Sets for Monomial Ideals 618 10. Hilbert Polynomial in the Affine Case 626 X. METHODS OF ALGEBRAIC GEOMETRY (Continued) 11. Hilbert Polynomial in the Projective Case 633 12. Intersections in Projective Space 635 13. Schemes 638 14. Problems 644 Hints for Solutions of 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 Spaces over Q, R, and C III. Inner-Product Spaces IV. Groups and Group Actions V. Theory of a Single Linear Transformation VI. Multilinear Algebra VII. Advanced Group Theory VIII. Commutative Rings and Their Modules IX. Fields and Galois Theory X. Modules over Noncommutative Rings