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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