Preface
Etymology
Special Notation
Chapter I Things Past
1.1. Some Number Theory
1.2. Roots of Unity
1.3. Some Set Theory
Chapter 2 Groups I
2.1. Introduction
2.2. Permutations
2.3. Groups
2.4. Lagranges Theorem
2.5. Homomorphisms
2.6. Quotient Groups
2.7. Group Actions
Chapter 3 Commutative Rings I
3.1. Introduction
3.2. First Properties
3.3. Polynomials
3.4. Greatest Common Divisors
3.5. Homomorphisms
3.6. Euclidean Rings
3.7. Linear Algebra
Vector Spaces
Linear Transformations
3.8. Quotient Rings and Finite Fields
Chapter 4 Fields
4.1. Insolvability of the Quintic
Formulas and Solvability by Radicals
Translation into Group Theory
4.2. Fundamental Theorem of Galois Theory
Chapter 5 Groups II
5.1. Finite Abelian Groups
Direct Sums
Basis Theorem
Fundamental Theorem
5.2. The Sylow Theorems
5.3. The Jordan-Hilder Theorem
5.4. Projective Unimodular Groups
5.5. Presentations
5.6. The Neilsen-Schreier Theorem
Chapter 6 Commutative Rings H
6.1. Prime Ideals and Maximal Ideals
6.2. Unique Factorization Domains
6.3. Noetherian Rings
6.4. Applications of Zoms Lemma
6.5. Varieties
6.6. Gr6bner Bases
Generalized Division Algorithm
Buchbergers Algorithm
Chapter 7 Modules and Categories
7.1. Modules
7.2. Categories
7.3. Functors
7.4. Free Modules, Projectives, and Injectives
7.5. Grothendieck Groups
7.6. Limits
Chapter 8 Algebras
8.1. Noncommutative Rings
8.2. Chain Conditions
8.3. Semisimple Rings
8.4. Tensor Products
8.5. Characters
8.6. Theorems of Burnside and of Frobenius
Contents
Chapter 9 Advanced Linear Algebra
9.1. Modules over PIDs
9.2. Rational Canonical Forms
9.3. Jordan Canonical Forms
9.4. Smith Normal Forms
9.5. Bilinear Forms
9.6. Graded Algebras
9.7. Division Algebras
9.8. Exterior Algebra
9.9. Determinants
9.10. Lie Algebras
Chapter 10 Homology
10.1. Introduction
10.2. Semidirect Products
10.3. General Extensions and Cohomology
10.4. Homology Functors
10.5. Derived Functors
10.6. Ext and Tor
10.7. Cohomology of Groups
10.8. Crossed Products
10.9. Introduction to Spectral Sequences
Chapter 11 Commutative Rings III
11.1. Local and Global
11.2. Dedekind Rings
Integrality
Nullstellensatz Redux
Algebraic Integers
Characterizations of Dedekind Rings
Finitely Generated Modules over Dedekind Rings
11.3. Global Dimension
11.4. Regular Local Rings
AppendixThe Axiom of Choice and Zorns Lemma
Bibliography
Index
This book can be used as a text for the first year of graduate algebra, but it is much morethan that. It can also serve more advanced graduate students wishing to learn topics ontheir own; while not reaching the frontiers, the book does provide a sense of the successesand methods arising in an area. Finally, this is a reference containing many of the standardtheorems and definitions that users of algebra need to know. Thus, the book is not only anappetizer, but a hearty meal as well.
缺书网
扫码进群