Abstract Algebra (2nd Edition) by Justin R. Smith

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Abstract Algebra (2nd Edition) written by Justin R. Smith. This book arose out of courses in Abstract Algebra, Galois Theory, Algebraic Geometry, and Manifold Theory the author taught at Drexel University. It is useful for self-study or as a course textbook. The first four chapters are suitable for the first of a two semester undergraduate course in Abstract Algebra and chapters five through seven are suitable for the second semester.

Abstract Algebra (2nd Edition) written by Justin R. Smith cover the following topics.

Foreword
List of Figures

1. Introduction

2. Preliminaries
2.1. Set theory
2.2. Operations on sets
2.3. The Power Set

3. A glimpse of number theory
3.1. Prime numbers and unique factorization
3.2. Modular arithmetic
3.3. The Euler f-function
3.4. Applications to cryptography

4. Group Theory
4.1. Introduction
4.2. Homomorphisms
4.3. Cyclic groups
4.4. Subgroups and cosets
4.5. Symmetric Groups
4.6. Abelian Groups
4.7. Group-actions
4.8. The Sylow Theorems
4.9. Subnormal series
4.10. Free Groups
4.11. Groups of small order

5. The Theory of Rings
5.1. Basic concepts
5.2. Homomorphisms and ideals
5.3. Integral domains and Euclidean Rings
5.4. Noetherian rings
5.5. Polynomial rings
5.6. Unique factorization domains

6. Modules and Vector Spaces
6.1. Introduction
6.2. Vector spaces
6.3. Modules
6.4. Rings and modules of fractions

7. Fields
7.1. Definitions
7.2. Algebraic extensions of fields
7.3. Computing Minimal polynomials
7.4. Algebraically closed fields
7.5. Finite fields
7.6. Transcendental extensions

8. Further topics in ring theory
8.1. Artinian rings
8.2. Integral extensions of rings
8.3. The Jacobson radical and Jacobson rings
8.4. Discrete valuation rings
8.5. Graded rings and modules

9. Galois Theory
9.1. Before Galois
9.2. Galois
9.3. Isomorphisms of fields
9.4. Roots of Unity
9.5. Cyclotomic Polynomials
9.6. Group characters
9.7. Galois Extensions
9.8. Solvability by radicals
9.9. Galois’s Great Theorem
9.10. The fundamental theorem of algebra

10. Division Algebras over R
10.1. The Cayley-Dickson Construction
10.2. Quaternions
10.3. Octonions and beyond

11. A taste of category theory
11.1. Introduction
11.2. Functors
11.3. Adjoint functors
11.4. Limits
11.5. Abelian categories
11.6. Tensor products
11.7. Tensor Algebras and variants

12. A little algebraic geometry
12.1. Introduction
12.2. Hilbert’s Nullstellensatz

13. Cohomology
13.1. Chain complexes and cohomology
13.2. Rings and modules
13.3. Cohomology of groups

Appendix
A. Axiomatic Set Theory
A.1. Introduction
A.2. Zermelo-Fraenkel Axioms
B. Solutions to Selected Exercises
Glossary
Index
Bibliography

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