Advanced algebra

Detalles Bibliográficos
Autor principal: Knapp, Anthony W.
Formato: Libro
Lenguaje:English
Publicado: Boston : Birkhäuser ; ©2007.
Colección:Cornerstones
Materias:
Teoría algebraica de los números
Tabla de Contenidos:
  • Of Basic Algebra x
  • Dependence among Chapters xvi
  • Guide for the Reader xvii
  • I 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
  • II 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 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
  • 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 Grobner Bases 491
  • 8 Constructive Existence 499
  • 9 Uniqueness of Reduced Grobner 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
  • 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.

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