Rosenberger G., Schürenberg A., Wienke L. Abstract Algebra: With Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography

3rd ed. — Berlin: De Gruyter, 2024. — 423 p. — (De Gruyter Textbook). — ISBN 9783111139514.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.We present the material sequentially, so that polynomials and field extensions precede an in-depth look at advanced topics in group theory and Galois theory. This text follows the new approach of conveying abstract algebra starting with rings and fields, rather than with groups. Our teaching experience shows that examples of groups seem rather abstract and require a certain formal framework and mathematical maturity that would distract a course from its main objectives. The idea is that the integers provide the most natural example of an algebraic structure that students know from school. A student who goes through ring theory first, will attain a solid background in abstract algebra and will be able to move on to more advanced topics.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.