Basic Algebra and Advanced Algebra

by Knapp, Anthony W.

• ISBN13: 9780817645335
• ISBN10: 0817645330
• Format: Hardcover
• Copyright: 2008-03-30
• Publisher: Birkhauser
• More Book Details

Summary

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole, presenting the subject matter in a forward-looking way that takes into account its historical development.Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry. The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them. The presentation includes blocks of problems that introduce additional topics and applications to science and engineering to guide further study. Many examples and hundreds of problems are included, along with separate sections giving hints or complete solutions for most of the problems.

Table of Contents

Contents of Basic Algebra	p. x
Preface	p. xi
List of Figures	p. xv
Dependence among Chapters	p. xvi
Guide for the Reader	p. xvii
Notation and Terminology	p. xxi
Transition to Modern Number Theory	p. 1
Historical Background	p. 1
Quadratic Reciprocity	p. 8
Equivalence and Reduction of Quadratic Forms	p. 12
Composition of Forms, Class Group	p. 24
Genera	p. 31
Quadratic Number Fields and Their Units	p. 35
Relationship of Quadratic Forms to Ideals	p. 38
Primes in the Progressions 4n + 1 and 4n + 3	p. 50
Dirichlet Series and Euler Products	p. 56
Dirichlet's Theorem on Primes in Arithmetic Progressions	p. 61
Problems	p. 67
Wedderburn-Artin Ring Theory	p. 76
Historical Motivation	p. 77
Semisimple Rings and Wedderburn's Theorem	p. 81
Rings with Chain Condition and Artin's Theorem	p. 87
Wedderburn-Artin Radical	p. 89
Wedderburn's Main Theorem	p. 94
Semisimplicity and Tensor Products	p. 104
Skolem-Noether Theorem	p. 111
Double Centralizer Theorem	p. 114
Wedderburn's Theorem about Finite Division Rings	p. 117
Frobenius's Theorem about Division Algebras over the Reals	p. 118
Problems	p. 120
Brauer Group	p. 123
Definition and Examples, Relative Brauer Group	p. 124
Factor Sets	p. 132
Crossed Products	p. 135
Hilbert's Theorem 90	p. 145
Digression on Cohomology of Groups	p. 147
Relative Brauer Group when the Galois Group Is Cyclic	p. 158
Problems	p. 162
Homological Algebra	p. 166
Overview	p. 167
Complexes and Additive Functors	p. 171
Long Exact Sequences	p. 184
Projectives and Injectives	p. 192
Derived Functors	p. 202
Long Exact Sequences of Derived Functors	p. 210
Ext and Tor	p. 223
Abelian Categories	p. 232
Problems	p. 250
Three Theorems in Algebraic Number Theory	p. 262
Setting	p. 262
Discriminant	p. 266
Dedekind Discriminant Theorem	p. 274
Cubic Number Fields as Examples	p. 279
Dirichlet Unit Theorem	p. 288
Finiteness of the Class Number	p. 298
Problems	p. 307
Reinterpretation with Adeles and Ideles	p. 313
p-adic Numbers	p. 314
Discrete Valuations	p. 320
Absolute Values	p. 331
Completions	p. 342
Hensel's Lemma	p. 349
Ramification Indices and Residue Class Degrees	p. 353
Special Features of Galois Extensions	p. 368
Different and Discriminant	p. 371
Global and Local Fields	p. 382
Adeles and Ideles	p. 388
Problems	p. 397
Infinite Field Extensions	p. 403
Nullstellensatz	p. 404
Transcendence Degree	p. 408
Separable and Purely Inseparable Extensions	p. 414
Krull Dimension	p. 423
Nonsingular and Singular Points	p. 428
Infinite Galois Groups	p. 434
Problems	p. 445
Background for Algebraic Geometry	p. 447
Historical Origins and Overview	p. 448
Resultant and Bezout's Theorem	p. 451
Projective Plane Curves	p. 456
Intersection Multiplicity for a Line with a Curve	p. 466
Intersection Multiplicity for Two Curves	p. 473
General Form of Bezout's Theorem for Plane Curves	p. 488
Grobner Bases	p. 491
Constructive Existence	p. 499
Uniqueness of Reduced Grobner Bases	p. 508
Simultaneous Systems of Polynomial Equations	p. 510
Problems	p. 516
The Number Theory of Algebraic Curves	p. 520
Historical Origins and Overview	p. 520
Divisors	p. 531
Genus	p. 534
Riemann-Roch Theorem	p. 540
Applications of the Riemann-Roch Theorem	p. 552
Problems	p. 554
Methods of Algebraic Geometry	p. 558
Affine Algebraic Sets and Affine Varieties	p. 559
Geometric Dimension	p. 563
Projective Algebraic Sets and Projective Varieties	p. 570
Rational Functions and Regular Functions	p. 579
Morphisms	p. 590
Rational Maps	p. 595
Zariski's Theorem about Nonsingular Points	p. 600
Classification Questions about Irreducible Curves	p. 604
Affine Algebraic Sets for Monomial Ideals	p. 618
Hilbert Polynomial in the Affine Case	p. 626
Hilbert Polynomial in the Projective Case	p. 633
Intersections in Projective Space	p. 635
Schemes	p. 638
Problems	p. 644
Hints for Solutions of Problems	p. 649
Selected References	p. 713
Index of Notation	p. 717
Index	p. 721
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