Contents of Basic Algebra ....................................... x
Preface ........................................................ xi
List of Figures ................................................ xv
Dependence among Chapters ..................................... xvi
Guide for the Reader ......................................... xvii
Notation and Terminology ...................................... xxi
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
10 Contents
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 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
Selected References ........................................... 713
Index of Notation ............................................. 117
Index ......................................................... 721
CONTENTS OF BASIC ALGEBRA
I Preliminaries about the Integers, Polynomials, and Matrices
II Vector Spaces over Q, R, and С
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
|
