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Abstract algebra is the study of algebraic structures like groups, rings and fields. This book provides an account of the theoretical foundations including applications to Galois Theory, Algebraic Geometry and Representation Theory. It implements the pedagogic approach to conveying algebra from the perspective of rings. The 3rd edition provides a revised and extended versions of the chapters on Algebraic Cryptography and Geometric Group Theory.
- Uses a modern approach to introduce algebra via rings and integers rather than via group theory.
- Covers unique topics such as Algebraic Geometry and cryptography.
- Includes important recent applications and accompanying exercises.
Preface
Groups, Rings and Fields
Maximal and Prime Ideals
Prime Elements and Unique Factorization Domains
Polynomials and Polynomial Rings
Field Extensions
Field Extensions and Compass and Straightedge Constructions
Kronecker’s Theorem and Algebraic Closures
Splitting Fields and Normal Extensions
Groups, Subgroups and Examples
Normal Subgroups, Factor Groups and Direct Products
Symmetric and Alternating Groups
Solvable Groups
Group Actions and the Sylow Theorems
Free Groups and Group Presentations
Finite Galois Extensions
Separable Field Extensions
Applications of Galois Theory
The Theory of Modules
Finitely Generated Abelian Groups
Integral and Transcendental Extensions
The Hilbert Basis Theorem and the Nullstellensatz
Algebras and Group Representations
Algebraic Cryptography
Non-Commutative Group Based Cryptography
Bibliography
Index